Spatial and Temporal Modulation of Thermoelectric Metamaterials

Abstract

In this chapter, we delve into the exciting realm of emerging multiphysics metamaterials, which holds great potential for orchestrating intricate physical processes. Our focus lies in exploring the concept of spatially and spatiotemporally modulated thermoelectric metamaterials, aiming to achieve multifaceted control over the thermal and electric domains. To establish a solid theoretical foundation for spatially modulated thermoelectric metamaterials, we delve into decoupled transformation thermoelectrics, coupled transformation thermoelectrics, and nonlinear transformation thermoelectrics. These theories serve as the basis for achieving functional control over electric and thermal fields. Recognizing the limitations of single-functionality and the inherent immutability of conventional thermoelectric metamaterials, a novel concept emerges-that of spatiotemporal multiphysics metamaterials, which incorporates the temporal dimension. This transformative facet empowers each physical field with manifold functionalities and the ability to seamlessly transition between them. To realize this vision, we develop rotatable checkerboard structures with varying rotation times, material compositions, and geometric shapes. These structures facilitate flexible switching of thermal and electric functions, enabling capabilities such as cloaking, sensing, and concentrating. The outcomes of our research offer a promising spatiotemporal platform for realizing adaptive and intelligent manipulation of multiphysics fields. This advancement paves the way for innovative applications and opens up new horizons in the field of metamaterials.

1 Opening Remarks

Whether in daily life or science and technology, the scenarios people face are always complex [1, 2]. The manipulation of intricate physical fields within complex scenarios plays a pivotal role across a broad spectrum of applications, spanning power generation [3], radiative cooling [4], biomedical engineering [5], and energy management [6]. Metamaterials [7,8,9] emerge as a formidable platform for orchestrating diverse physical domains encompassing optics [10], plasma physics [11], acoustics [12, 13], thermotics [14,15,16,17,18,19,20,21,22], electrics [23, 24], and fluidics [25, 26]. Exciting functions like cloaking, sensing, and concentrating allow for the physical field intensity in the working region to be far smaller, equal, or larger than in the background [27,28,29,30]. As the demand for multifaceted applications escalates, the evolution of metamaterial design transcends the confines of single-field manipulation. Multiphysics metamaterials emerge as a beacon of promise, heralding the potential to govern multiple physical domains in concert. This transformative trajectory gives rise to the simultaneous governance of electromagnetic and acoustic fields [31, 32], conductive and convective or radiative domains [33,34,35,36], and the nuanced interplay of thermal and electric realms [37,38,39,40,41,42]. The intertwining of thermal and electrical transport, often intertwined, spawns a quintessential multi-physics paradigm. The employment of transformation theory to craft thermoelectric metamaterials provides a potent avenue to realize spatial modulation within the thermal and electric landscapes. Expanded theories, including effective medium and scattering cancellation theories, furnish blueprints for the tangible realization of functional thermal and electric domains. However, extant multiphysics metamaterials are not without their limitations. Traditional methodologies proffer fixed functionalities owing to unyielding structures and materials. The quest for continuous controllability is met with tribulation, as it necessitates concurrent and on-demand adjustments to multiphysical parameters. Thus, the pursuit of multifunctionality within each physical domain, not to mention its seamless toggling, remains an imposing challenge within the realm of multiphysics metamaterials.

Recently, metamaterial designs incorporating the temporal dimension have garnered considerable attention due to their promising degree of freedom [43, 44]. This augmentation with spatiotemporal modulation not only begets unforeseen nonreciprocal phenomena [45,46,47,48,49,50] and captivating topological transport [51,52,53] from a fundamental physics perspective but also finds pragmatic application in adaptive thermal camouflage [54] and real-time digital coding [55, 56]. However, the current scope of spatiotemporal strategies is predominantly constrained to single physical fields, and extending their purview to multiphysics domains presents a formidable conundrum, entailing intricate coordination of parameters across diverse physical realms, all while adhering to the constraints of materials and structures.

In this chapter, we introduce the field of spatially regulated thermoelectric metamaterials based on transformation thermoelectricity and provide a summary of the existing functionalities. Furthermore, we unravel the conceptual framework of spatiotemporal multiphysics metamaterials, which represents an innovative approach enabling simultaneous manipulation of thermal and electric fields [57]. To incorporate the temporal dimension, we employ a rotatable checkerboard structure that facilitates continuous tuning of its geometric configuration over time. This characteristic grants us flexible control over thermal and electric conductivities, thus enabling the realization of various functions in both fields. Our spatiotemporal multiphysics metamaterials offer the flexibility to switch between three or five function combinations, depending on the number of constituent materials used. These findings unlock novel possibilities for flexible and intelligent control of multiphysics fields, presenting opportunities for advanced applications.

2 Space-Regulated Thermoelectric Metamaterials

Thermoelectric devices are pivotal in both industrial applications and daily life, necessitating the control of thermal and electric fields within a single unit. Within the thermoelectric field, the thermal and electric fields can either operate independently (decoupled) or in conjunction (coupled). In a decoupled scenario, a device demonstrating temperature and voltage differences manages its thermal and electric fields independently, with each field unaffected by the other. On the other hand, in a coupled setup, the thermal and electric fields influence each other, rendering their separate control unfeasible. Thus, when considering the thermoelectric effect, the coupling term becomes indispensable, adding a layer of complexity to the analysis of thermoelectric metamaterials.

2.1 Decoupled Transformation Thermoelectrics

First, we consider the transformation theory within decoupled thermoelectric fields. In such fields, the steady-state heat conduction equation without a heat source parallels the electrical conduction equation without a power supply in form, both of which can be expressed as

$$\begin{aligned} \nabla \cdot (\kappa \nabla T)=0, ~~ \nabla \cdot (\sigma \nabla \mu )=0, \end{aligned}$$

(6.1)

where $\kappa$ and $\sigma$ are thermal conductivity tensor and electrical conductivity tensor, T and $\mu$ are temperature and potential. Li et al. proved that the Eq. (6.1) satisfies formal invariance under coordinate transformation, and the transformation rule is [37]

$$\begin{aligned} \kappa '=\frac{\textsf{A}\kappa _{0}\textsf{A}^{tr}}{\det \textsf{A}}, ~~\sigma '=\frac{\textsf{A}\sigma _{0}\textsf{A}^{tr}}{\det \textsf{A}}, \end{aligned}$$

(6.2)

where $\kappa '$ and $\sigma '$ are transformed thermal conductivity and electric conductivity, $\kappa _{0}$ and $\sigma _{0}$ are thermal conductivity and electric conductivity of the background, $\textsf{A}$ is the Jacobian transformation matrix. The physical parameters of real space can be obtained by substituting coordinate transformation into the Jacobi matrix.

2.2 Coupled Transformation Thermoelectrics

In coupled thermoelectric fields, a mutual conversion between electricity and heat exists. The Peltier effect refers to the generation of a heat flow due to a potential difference. As charge carriers flow from higher to lower energy levels, surplus heat is discharged into the material. Conversely, when charge carriers move inversely, heat is absorbed, leading to the material’s cooling. The Seebeck effect, on the other hand, describes the current induced by a temperature gradient. Here, charge carriers move from the hot to the cold end, accumulating at the latter. This buildup results in an electric potential difference, subsequently inducing an electric current in the reverse direction. The Seebeck coefficient captures the strength of this interaction, coupling the thermal and electric fields. When a temperature and voltage difference coexist, coupled heat and current emerge alongside their independent movements. In the linear regime, the coupling between the electric, $\boldsymbol{J}_E$, and the heat current density, $\boldsymbol{J}_Q$, is expressed as

$$\begin{aligned} \begin{aligned} {\boldsymbol{J}}_E&=-\sigma \nabla \mu - \sigma S \nabla T,\\ {\boldsymbol{J}}_Q&=-\kappa \nabla T+T S^{\dag }{\boldsymbol{J}}_E, \end{aligned} \end{aligned}$$

(6.3)

where $\kappa$ and $\sigma$ are thermal and electrical conductivity tensors, respectively. T is temperature, $\mu$ is the electric potential, S is the Seebeck coefficient tensor, and $S^{\dag }$ is its transpose. When considering the steady state with local equilibrium and the absence of external heat and power sources, the governing equation for the thermoelectric effect can be expressed as follows

$$\begin{aligned} \begin{aligned} \nabla \cdot \boldsymbol{J}_E&=0,\\ \nabla \cdot \boldsymbol{J}_Q&=-\nabla \mu \cdot \boldsymbol{J}_E. \end{aligned} \end{aligned}$$

(6.4)

These two equation express the local conservation in the steady state of charge and energy, respectively. The thermoelectric coupling transport produces a heat source term, $-\nabla \mu \cdot {\boldsymbol{J}}_E$, corresponding to the Joule effect. In view of Onsager reciprocal relations, electric and thermal conductivity matrices should be symmetric, $\sigma =\sigma ^\dag$ and $\kappa = \kappa ^{\dag }$. Substituting Eq. (6.3) into Eq. (6.4), the complete governing equation for the thermoelectric effect can be formulated as

$$\begin{aligned} \begin{aligned} \nabla \cdot \left( \sigma \nabla \mu +\sigma S\nabla T\right) &=0 \end{aligned} \end{aligned}$$

(6.5)

and

$$\begin{aligned} \begin{aligned} -\nabla \cdot \left( \kappa \nabla T+ T S^{\dag }\sigma S\nabla T + T S^{\dag } \sigma \nabla \mu \right) &=\nabla \mu \cdot \left( \sigma \nabla \mu +\sigma S\nabla T\right) . \end{aligned} \end{aligned}$$

(6.6)

Next, it needs to be proved that the governing equations (6.5) and (6.6) remain invariant under any coordinate transformation. In a curvilinear coordinate system characterized by contravariant bases ${\boldsymbol{g}^i,\boldsymbol{g}^j,\boldsymbol{g}^k}$, covariant bases ${\boldsymbol{g}_i,\boldsymbol{g}_j,\boldsymbol{g}_k}$, and their respective contravariant components $(x^i,y^j,z^k)$, the heat and electrical conduction terms on the left side of Eq. (6.5) can be reformulated as

$$\begin{aligned} \begin{aligned} \nabla \cdot \left( \sigma \nabla \mu +\sigma S\nabla T\right) =\frac{1}{\sqrt{g}}(\partial _i\sqrt{g}\sigma ^{ij}\partial _j\mu +\partial _i\sqrt{g}\sigma ^{ij}S_j^k\partial _k T), \end{aligned} \end{aligned}$$

(6.7)

where g is the determinant of the matrix with components $g_{ij}={\boldsymbol{g}}_i\cdot {\boldsymbol{g}}_j$. Hence, the component form of Eq. (6.5) can be expressed as

$$\begin{aligned} \partial _i(\sqrt{g}\sigma ^{ij}\partial _j\mu )+\partial _i(\sqrt{g}\sigma ^{ij}S_j^k\partial _k T)=0. \end{aligned}$$

(6.8)

According to Eq. (6.8), the form invariance of Eq. (6.5) is verified. Following the above deduction, the component form of Eq. (6.6) can be written. The right-hand term of Eq. (6.6) can be written as

$$\begin{aligned} \begin{aligned} \nabla \mu \cdot \left( \sigma \nabla \mu +\sigma S\nabla T\right) =(\partial _i\mu )(\sigma ^{ij}\partial _j\mu +\sigma ^{ij}S_j^k\partial _k T). \end{aligned} \end{aligned}$$

(6.9)

Then the component form of each term on the left side of the Eq. (6.6) is written as

$$\begin{aligned} \begin{aligned} -\nabla \cdot \left( \kappa \nabla T\right) &=-\partial _j[\sqrt{g}\kappa ^{jk}\partial _k T]/\sqrt{g};\\ -\nabla \cdot \left( T S^{\dag }\sigma S\nabla T\right) &=-\partial _j[\sqrt{g}T(S^{\textrm{tr}})_i^j\sigma ^{ij}S_j^k\partial _k{T}]/\sqrt{g};\\ -\nabla \cdot \left( T S^{\dag } \sigma \nabla \mu \right) &=-\partial _j[\sqrt{g}T(S^{\textrm{tr}})_i^j\sigma ^{ij}\partial _j{\mu }]/\sqrt{g}, \end{aligned} \end{aligned}$$

(6.10)

where $(S^{\textrm{tr}})_i^j$ is the transpose of $S_i^j$. Substituting Eqs. (6.9) and (6.10) into Eq. (6.6), the result can be written as

$$\begin{aligned} \begin{aligned} \left. \begin{array}{c} \partial _j[\sqrt{g}(\kappa ^{jk}\partial _k{T}+T(S^{\textrm{tr}})_i^j\sigma ^{ij}S_j^k \partial _k{T}+T(S^{\textrm{tr}})_i^j\sigma ^{ij}\partial _j\mu )]\\ =-\sqrt{g}(\partial _i\mu )(\sigma ^{ij}\partial _j\mu +\sigma ^{ij}S_j^k\partial _k{T}). \end{array} \right. \end{aligned} \end{aligned}$$

(6.11)

Similarly, Eq. (6.11) maintains its form across various coordinates, akin to Eq. (6.8). With this, the form invariance of the coupled thermoelectric governing equations under coordinate transformations has confirmed in Eqs. (6.8) and (6.11). The sole distinction across different coordinate systems is the coefficient g. As a result, transformation theory can be applied to the coupled thermoelectric field to regulate the linearly coupled heat charge flow.

To obtain transformation rules for material properties, the approach shifts from coordinate transformation to geometric mapping. Consider a bijection $f: r\mapsto r'$ in the three-dimensional Euclidean space that smoothly maps the pre-transformed space to the transformed space. Owing to the diffeomorphism between the pre-transformed space (virtual space) with curvilinear coordinates $\{x^i,x^j,x^k\}$ and the transformed space (physical space) with Cartesian coordinates $\{x',y',z'\}$, the temperature and electrochemical potential in Eqs. (6.5) and (6.6) are replaced by $T'( r')=T'(x',y',z')$ and $\mu '( r')=\mu '(x',y',z')$ if $T'( r')=T(f^{-1}( r'))$ and $\mu '( r')=\mu (f^{-1}( r'))$. They still satisfy

$$\begin{aligned} \nabla ' \cdot (\sigma '\nabla '\mu ' + \sigma 'S'\nabla ' T')=0, \end{aligned}$$

(6.12)

and

$$\begin{aligned} \begin{aligned} \left. \begin{array}{c} -\nabla ' \cdot (\kappa '\nabla ' T'+T'(S')^{\textrm{tr} } \sigma ' S'\nabla ' T'+T'(S')^{\textrm{tr} }\sigma '\nabla '\mu ')\\ =\nabla '\mu ' \cdot (\sigma '\nabla '\mu ' + \sigma ' S'\nabla ' T'). \end{array} \right. \end{aligned} \end{aligned}$$

(6.13)

Comparing Eqs. (6.8) and (6.12), (6.11) and (6.13), the transformed thermal conductivity $\kappa '$, electrical conductivity $\sigma '$, and Seebeck coefficient $S'$ in the coupled thermoelectric field can be written as

$$\begin{aligned} \begin{aligned} \kappa '(r')=&\frac{\textsf{A}\kappa {\textsf{A}}^{\textrm{tr} }}{\det {\textsf{A}}}\\ \sigma '(r')=&\frac{\textsf{A}\sigma {\textsf{A}}^{\textrm{tr} }}{\det {\textsf{A}}}\\ S'(r')=&{\textsf{A}}^{-\textrm{tr} }S{\textsf{A}}^{\textrm{tr} }. \end{aligned} \end{aligned}$$

(6.14)

By designing the coordinate transformation and substituting it into the Jacobian matrix to obtain the transformed rules, the spatial function adjustment of the thermal field and electric field can be realized.

2.3 Temperature-Dependent Transformation Thermoelectrics

Nonlinear materials open new avenues in material design and functional deployment. Introducing nonlinearity in thermal metamaterial designs expands the scope for modifying thermal field functions. Transitioning from linear to nonlinear transformation theories across diverse physical fields enhances flexibility and adaptability in complex real-world scenarios. In the nonlinear approach, one can either consider transformations that are temperature-dependent or independent, juxtaposed with temperature-dependent background parameters. In the realm of the thermoelectric effect, characterized by the Seebeck coefficient, the structure of the governing equation aligns with Eq. (6.3). It considers the temperature-dependent thermal conductivity $\kappa (T)$, electrical conductivity $\sigma (T)$, and Seebeck coefficient S(T). Notably, even with temperature changes, these equations maintain their structural integrity during coordinate changes, giving rise to the concept of temperature-dependent transformation thermoelectricity [58]. As per this theory, the transformation rules manifest as

$$\begin{aligned} \begin{aligned} \kappa '(T')=&\frac{\textsf{A}\kappa (T){\textsf{A}}^{\textrm{tr} }}{\det {\textsf{A}}}\\ \sigma '(T')=&\frac{\textsf{A}\sigma (T){\textsf{A}}^{\textrm{tr} }}{\det {\textsf{A}}}\\ S'(T')=&{\textsf{A}}^{-\textrm{tr} }S(T){\textsf{A}}^{\textrm{tr} }. \end{aligned} \end{aligned}$$

(6.15)

where $\kappa (T)$, $\sigma (T)$, S(T) are the pre-transformation parameters that are the temperature-dependent background parameters. The transformed parameters $\kappa '(T')$, $\sigma '(T')$, $S'(T')$ are related to temperature and the transformed coordinates. The temperature-dependent transformation theory also applies to the decoupling thermoelectric field, corresponding to the coupled thermoelectric field at $S(T)=0$.

In essence, both decoupled and coupled thermoelectric fields maintain their structural integrity in coordinate transformations, irrespective of the temperature sensitivity of the background parameters. This means transformation theories, both temperature-dependent and independent, apply seamlessly to these fields. Thermal and electric fields can be designed to achieve specific tasks by using coordinate transformations. For example, cloak transformations can be embedded into Eqs. (6.2), (6.14), and (6.15) to manifest thermal and electric cloaking, granting dual-field control in diverse settings. Transformation theory thus paves the way for a multitude of functionalities, such as sensors, concentrators, rotators, and camouflages.

2.4 Functional Realization of Thermal and Electric Fields

In practical applications, the search for anisotropic thermoelectric materials required by transformation theory poses a challenge. Therefore, in the realm of thermoelectric metamaterials, the effective medium theory and scattering cancellation theory are commonly employed. To begin with, let’s discuss the functionality of decoupled thermoelectric fields. An experimental realization of a thermal and electric concentrator was achieved by employing the effective medium theory [41]. In this proposed configuration, the concentrator shell comprises wedges made of two different materials, arranged alternately to form an annulus. Similarly, different functions can be attained for electric and thermal fields. Simultaneous thermal concentration and electric cloaking were accomplished by embedding a mixture of various shapes and materials within the host medium [39]. Additionally, an invisible sensor capable of sensing and cloaking thermal and electric fields simultaneously was proposed and realized [40]. Rich composite materials bring new life to functional designs. Controlling colloidal particles in soft matter enables the control of the properties of electric and magnetic fields [59,60,61]. Similarly, composite thermal cloak shells can be realized by distributing non-spherical nanoparticles of different shapes and volume fractions along the radius of the cloak [37]. The effective electric and thermal conductivity, calculated using the effective medium theory, was found to satisfy the cloaking requirements derived from transformation theory [37]. Furthermore, thermal and electric cloaks were experimentally realized in a bilayer structure designed based on scattering cancellation theory [38].

Fig. 6.1

Multifunctional devices for decoupled thermoelectric fields. Concept of spatiotemporal multiphysics metamaterials. The two-material-based and four-material-based checkerboard structures are shown in a and b, respectively. By rotating the even-numbered layers at a constant angular speed, the heat and electric currents in the central region can be altered, allowing for continuous and seamless function adjustment. (from Ref. [57], licensed under CC-BY 4.0)

Transformation theory also provides insights into achieving fundamental functions such as cloaking, concentration, and rotation in coupled thermoelectric fields [58, 62]. To address the challenge of preparing coupled thermoelectric metamaterials with parameters required by transformation theory, a bilayer structure based on the generalized scattering-cancellation method has been developed, effectively reducing the difficulty [63]. Practical applications in coupled thermoelectric fields have been proposed, leveraging the temperature-dependent transformation theory [58]. Firstly, a switchable thermoelectric device has been introduced, capable of adapting to ambient temperature by switching between cloak and concentrator modes. Secondly, an improved thermoelectric cloak has been devised to maintain a nearly constant internal temperature. In coupled thermoelectric fields, independent control of thermal and electric fields is challenging due to coupling terms, limiting the flexible control of functions. However, this challenge can be overcome through a general computational scheme that designs composite materials based on the principles of simple circuit theory, enabling enhanced or suppressed current and thermal current in the desired direction [64].

3 Spatiotemporal Thermoelectric Metamaterials

3.1 Spatiotemporal Efficient Medium Theory

For enhanced functionality in thermoelectric metamaterials, it’s imperative to explore strategies beyond just the temperature-dependent transformation theory. A notable approach focuses on tweaking the geometric configurations of the metamaterials. Yet, managing multiple physical fields in tandem is complex, primarily due to the intricacies of adjusting various parameters simultaneously. To address this, spatiotemporal multiphysics metamaterials are introduced. This approach extends the available degrees of freedom, facilitating more precise functional adjustments [57]. The ambition is to craft a singular device, versatile in housing numerous functions for both thermal and electric fields, whilst permitting ongoing function switching. A typical combination in metamaterial functionality is the simultaneous realization of thermal cloaking and electric concentration. This dual capability is achievable through spatiotemporal multiphysics metamaterials designed as rotatable checkerboards. As illustrated in Fig. 6.1a, b, these checkerboard configurations incorporate either two or four isotropic materials. These materials play a pivotal role in dictating the potential function combinations for thermal and electric fields. By systematically rotating the even-numbered layers of the checkerboard, one can achieve dynamic functional adaptability. The inherent geometry of the structure defines the specific functions exhibited during this rotation. By expertly controlling the spatial arrangement and rotation time of the checkerboard components, one can finely tune both thermal and electric fields.

The checkerboard’s effective thermal and electric conductivities determines the thermal and electric currents in the central sector. Building upon the Keller theorem [65] and the effective medium theory [66], the checkerboard’s effective thermal and electrical conductivities in polar coordinates are given by

$$\begin{aligned} \kappa _r \kappa _\theta =\kappa _1\kappa _2, ~~~\sigma _r\sigma _\theta =\sigma _1\sigma _2, \end{aligned}$$

(6.16)

where $\kappa _r$ ($\sigma _r$) and $\kappa _{\theta }$ ($\sigma _{\theta }$) represent the effective radial and tangential thermal (electric) conductivities. $\kappa _1$ ($\sigma _1$) and $\kappa _2$ ($\sigma _2$) denote the thermal (electric) conductivities of the two materials constituting the checkerboard. It’s to note that $\kappa _{1}\kappa _{2}=\kappa _{r}\kappa _{\theta }=\kappa _b^2$ ($\sigma _{1}\sigma _{2}=\sigma _{r}\sigma _{\theta }=\sigma _b^2$), meaning the effective product of the radial and tangential thermal (electrical) conductivities equals the square of the background thermal (electrical) conductivity. This ensures that the background heat (electric) flow remains undisturbed. The ratio $\kappa _{r}/\kappa _{\theta }$ ($\sigma _{r}/\sigma _{\theta }$) dictates the heat (electric) flow distribution within the central region.

If $\kappa _{r}/\kappa _{\theta }<1$ ($\sigma _{r}/\sigma _{\theta }<1$), heat (electric) flow diverts around the central zone, leading to thermal (electric) cloaking. A ratio of zero produces an ideal cloaking effect. For $\kappa _{r}/\kappa _{\theta }=1$ ($\sigma _{r}\sigma _{\theta }=1$), the heat (electric) flow remains consistent within the center, resulting in thermal (electric) sensing. When $\kappa _{r}/\kappa _{\theta }>1$ ($\sigma _{r}/\sigma _{\theta }>1$), heat (electric) flow accumulates in the central region, achieving thermal (electric) concentration. Thus, forecasting the spatiotemporal function primarily hinges on determining the effective radial and tangential thermal (electric) conductivities of the checkerboard design.

Fig. 6.2

Schematic diagram of partial structure of space-time thermoelectric metamaterial. a A checkerboard structure composed of two materials. Brown arrows depict the rotation direction of even layers. b An expanded version of the structure from a, showcasing a multilayered configuration. c Individual presentation of thermal and electric conductivities. d Theoretical analyses correlating to the structure described in a. e–h Structures analogous to a–d, with the key difference being that the checkerboard design incorporates four materials. Functions underscored by green triangles denote sensing. (from Ref. [57], licensed under CC-BY 4.0)

A checkerboard structure can be composed of either two or four distinct materials. Figure 6.2a depicts an exemplary configuration utilizing two isotropic materials, each characterized by constant thermal ($\kappa$) and electrical ($\sigma$) conductivities. In this structure, even-numbered layers rotate clockwise while the odd-numbered layers remain stationary. The fixed boundary is highlighted by a red dashed line in Fig. 6.2a, with the evolving boundary indicated by a rotating brown dashed line. The term $\Delta \theta$ represents the central angle for a single phase unit, and $\tau$ denotes the duration needed for the even-numbered layers to complete a rotation of $\Delta \theta$. The angular displacement, represented by $\theta (t)$, follows the equation $\theta (t)=\Delta \theta t/\tau$, where t indicates the time elapsed. After a rotation spanning $2\tau$, the checkerboard returns to its initial position. Owing to this periodic behavior, the analysis focuses on one rotational cycle. By modulating the angle $\theta (t)$, the sector associated with the central angle $\Delta \theta$ can be segmented into areas displaying both uniform and alternating phases.

Within the rotatable checkerboard design, electric and thermal conductivities can be modulated independently, facilitating separate control of electric and thermal flows. This is evident from Fig. 6.2c. Due to the analogous variations in thermal and electric conductivities in the two-material checkerboard setup, theoretical analysis predominantly centers on thermal conductivity. The subsequent analysis for effective electric conductivity is achieved by substituting $\kappa _1$ and $\kappa _2$ with $\sigma _1$ and $\sigma _2$.

The homogeneous phase is characterized by units made of the same materials, with thermal and electrical conductivities designated as either $(\kappa _1, \sigma _1)$ or $(\kappa _2, \sigma _2)$. Conversely, the staggered phase integrates two interleaved isotropic material units. Over the duration $2\tau$, the central angle corresponding to the staggered phases is defined as

$$\begin{aligned} \theta _{s}(t)=\left\{ \begin{array}{cc} \frac{\Delta \theta }{\tau }t, &{} 0<t<\tau \\ 2\Delta \theta -\frac{\Delta \theta }{\tau }t. &{} \tau <t<2\tau \end{array} \right. \end{aligned}$$

(6.17)

In the timeframe $0<t<\tau$, the angle $\theta _{s}(t)$ expands, whereas in $\tau <t<2\tau$, it contracts. This results in the checkerboard configuration reverting to its original state at $2\tau$, devoid of any staggered phase. The effective thermal conductivity of these staggered phases is determined by both the material and geometric attributes of their building blocks [67]. The staggered phases’ geometric parameter evolves with time and is articulated as $\eta (t)=\ln (r_{i+1}/r_i)/\theta _{s}(t)$, where $r_i$ signifies the radius of the ith layer. And the shape parameter $\eta =\ln (r_{i+1}/r_i)/\Delta \theta$ is dictated by the geometric attributes of the checkerboard structure and is time-independent. In polar coordinates, the effective thermal conductivity $\kappa _c$ of the staggered phases can be divided into radial thermal conductivity $\kappa _{cr}$ and tangential thermal conductivity $\kappa _{c\theta }$, respectively,

$$\begin{aligned} \begin{array}{c} \kappa _{cr}=\left\{ \begin{array}{cc} \dfrac{\sqrt{\kappa _1 \kappa _2}}{(1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{t}{\eta \tau }-1)+1}, &{}0\le t \le \eta \tau \\ \sqrt{\kappa _1 \kappa _2}\left[ (1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{\eta \tau }{t}-1)+1\right] , &{} \eta \tau \le t \le \tau \\[2ex] \sqrt{\kappa _1 \kappa _2}\left[ (1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{\eta \tau }{2\tau -t}-1)+1\right] , &{}\tau \le t \le (2-\eta ) \tau \\[2ex] \dfrac{\sqrt{\kappa _1 \kappa _2}}{(1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{2\tau -t}{\eta \tau }-1)+1}, &{} (2-\eta ) \tau \le t \le 2\tau \\[2ex] \end{array} \right. \\ \\ \kappa _{c\theta }=\left\{ \begin{array}{cc} \sqrt{\kappa _1 \kappa _2}\left[ (1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{t}{\eta \tau }-1)+1\right] , &{}0 \le t \le \eta \tau \\[2ex] \dfrac{\sqrt{\kappa _1 \kappa _2}}{(1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{\eta \tau }{t}-1)+1}, &{} \eta \tau \le t \le \tau \\[2ex] \dfrac{\sqrt{\kappa _1 \kappa _2}}{(1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{\eta \tau }{2\tau -t}-1)+1}, &{} \tau \le t \le (2-\eta ) \tau \\[2ex] \sqrt{\kappa _1 \kappa _2}\left[ (1-\dfrac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\dfrac{2\tau -t}{\eta \tau }-1)+1\right] . &{} (2-\eta ) \tau \le t \le 2\tau \end{array} \right. \end{array} \end{aligned}$$

(6.18)

Throughout the entire $2\tau$ duration, the effective radial and tangential thermal conductivities of the staggered phase are defined by segmentation functions, displaying a temporal variation. The effective thermal conductivity of the interleaved phase is closely tied to the checkerboard unit’s shape parameter, $\eta$, with outcomes from Eq. (6.18) applying for $\eta \le 1$. For instances where $\eta >1$, the contradictory time range in the piecewise function of Eq. (6.18) can be omitted. It’s noteworthy that the staggered phase’s thermal conductivity, represented as $\kappa _c$, shows distinct values in the radial versus tangential directions.

The transformation of the fan-shaped configuration in Fig. 6.2a into a multi-layer arrangement as depicted in Fig. 6.2b involves the incorporation of staggered and homogeneous phases. The lateral thermal conductivity of this multi-layer structure corresponds to the tangential thermal conductivity of the initial fan-shaped design, while the longitudinal thermal conductivity aligns with the radial thermal conductivity of the original structure. Employing a stepwise application of the effective medium theory enables an approximation of the multi-layer structure, thus yielding the effective radial and tangential thermal conductivities for the rotatable checkerboard arrangement.

A strategic approach combines staggered and homogeneous phases to form a core unit. As a result, two distinct combined units, labeled as unit A and unit B, are defined: the former pairs $\kappa _c$ with $\kappa _1$, and the latter associates $\kappa _c$ with $\kappa _2$. Leveraging the effective medium theory, the radial and tangential effective thermal conductivity for these units can be derived as

$$\begin{aligned} \begin{array}{c} \kappa _{Ar}=\frac{\theta _s(t)}{\Delta \theta }\kappa _{cr}+\frac{\Delta \theta -\theta _s(t)}{\Delta \theta }\kappa _2, \kappa _{Br}=\frac{\theta _s(t)}{\Delta \theta }\kappa _{cr}+\frac{\Delta \theta -\theta _s(t)}{\Delta \theta }\kappa _1.\\ \\ \frac{1}{\kappa _{A\theta }}=\frac{\theta _s(t)}{\Delta \theta } \frac{1}{\kappa _{c\theta }}+\frac{\Delta \theta -\theta _s(t)}{\Delta \theta }\frac{1}{\kappa _2}, \frac{1}{\kappa _{B\theta }}=\frac{\theta _s(t)}{\Delta \theta }\frac{1}{\kappa _{c\theta }}+\frac{\Delta \theta -\theta _s(t)}{\Delta \theta }\frac{1}{\kappa _1}. \end{array} \end{aligned}$$

(6.19)

where $\theta _s(t)/\Delta \theta$ represents the staggered phase’s proportion within a unit, while $(\Delta \theta -\theta _s(t))/\Delta \theta$ is indicative of the homogeneous phase. With this understanding, the entire structure can be visualized as a multi-layered arrangement with alternating units. By applying the effective medium theory

$$\begin{aligned} \kappa _r=\frac{\kappa _{Ar}+\kappa _{Br}}{2},~\frac{1}{\kappa _\theta }=\frac{1}{2\kappa _{A\theta }}+\frac{1}{2\kappa _{B\theta }}. \end{aligned}$$

(6.20)

Both units A and B in the checkerboard layout share equivalent central angles. By solving Eqs.(6.18)–(6.20), the radial and tangential effective thermal conductivities for a rotatable checkerboard structure with two isotropic materials can be expressed as

$$\begin{aligned} \begin{array}{cc} \kappa _{r}=\left\{ \begin{array}{cc} \displaystyle \frac{t}{\tau }\frac{\sqrt{\kappa _1 \kappa _2}}{(1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{t}{\eta \tau }-1)+1}+(\frac{1}{2}-\frac{t}{2\tau })(\kappa _1+\kappa _2), &{}0 \le t \le \eta \tau \\[2ex] \displaystyle \frac{t}{\tau }\sqrt{\kappa _1 \kappa _2}\left[ (1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\eta \frac{\tau }{t}-1)+1\right] +(\frac{1}{2}-\frac{t}{2\tau })(\kappa _1+\kappa _2), &{}\eta \tau \le t \le \tau \\[2ex] \displaystyle (2-\frac{t}{\tau }) \sqrt{\kappa _1 \kappa _2}\left[ (1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{\eta \tau }{2\tau -t}-1)+1\right] +(\frac{t}{2\tau }-\frac{1}{2})(\kappa _1+\kappa _2), &{}\tau \le t \le (2-\eta ) \tau \\[2ex] \displaystyle (2-\frac{t}{\tau }) \frac{\sqrt{\kappa _1 \kappa _2}}{(1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{2\tau -t}{\eta \tau }-1)+1}+(\frac{t}{2\tau }-\frac{1}{2})(\kappa _1+\kappa _2), &{}(2-\eta ) \tau \le t \le 2\tau \\[2ex] \end{array} \right. \\ \\ \kappa _{\theta }=\left\{ \begin{array}{cc} \displaystyle \frac{\kappa _1\kappa _2}{\frac{t}{\tau }\frac{\sqrt{\kappa _1 \kappa _2}}{(1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{t}{\eta \tau }-1)+1}+(\frac{1}{2}-\frac{t}{2\tau })(\kappa _1+\kappa _2)}, &{}0 \le t \le \eta \tau \\[3ex] \displaystyle \frac{\kappa _1\kappa _2}{\sqrt{\kappa _1\kappa _2}\frac{t}{\tau }\left[ (1-\frac{2\sqrt{\kappa _1\kappa _2}}{\kappa _1+\kappa _2})(\frac{\eta \tau }{t}-1)+1\right] +(\frac{1}{2}-\frac{t}{2\tau })(\kappa _1+\kappa _2)}, &{}\eta \tau \le t \le \tau \\[3ex] \displaystyle \frac{\kappa _1\kappa _2}{(2-\frac{t}{\tau }) \sqrt{\kappa _1 \kappa _2}\left[ (1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{\eta \tau }{2\tau -t}-1)+1\right] +(\frac{t}{2\tau }-\frac{1}{2})(\kappa _1+\kappa _2)}, &{}\tau \le t \le (2-\eta ) \tau \\[3ex] \displaystyle \frac{\kappa _1\kappa _2}{(2-\frac{t}{\tau }) \frac{\sqrt{\kappa _1 \kappa _2}}{(1-\frac{2\sqrt{\kappa _1 \kappa _2}}{\kappa _1+\kappa _2})(\frac{2\tau -t}{\eta \tau }-1)+1}+(\frac{t}{2\tau }-\frac{1}{2})(\kappa _1+\kappa _2)}. &{} (2-\eta ) \tau \le t \le 2\tau \end{array} \right. \end{array} \end{aligned}$$

(6.21)

Replacing $\kappa _1$ and $\kappa _2$ with $\sigma _1$ and $\sigma _2$ can obtain the radial and tangential effective conductivities of the rotatable checkerboard structure composed of two isotropic materials.

Figure 6.2d describes the time variation of the effective thermal conductivity and electrical conductivity of the rotatable checkerboard structure within a single period $2\tau$, where $\tau =4$ s. This checkerboard design consists of two materials characterized by parameters ($\kappa _1=20 ~{{\textrm{W m}}^{-1} {\textrm{K}}^{-1}}$, $\sigma _1=20~{{\textrm{S m}}^{-1}}$) and ($\kappa _2=0.05~ {{\textrm{W m}}^{-1} {\textrm{K}}^{-1}}$, $\sigma _2=0.05~{{\textrm{S m}}^{-1}}$). Furthermore, the background and central regions exhibit thermal and electric conductivities of $\kappa _b=1~ {{\textrm{W m}}^{-1} {\textrm{K}}^{-1}}$ and $\sigma _b=1~{{\textrm{S m}}^{-1}}$, respectively. The outcomes displayed in Fig. 6.2d are computed utilizing Eq. (6.21). In the figure, the thermal and electric conductivities are represented by dashed and solid lines, respectively. The convergence of these two lines signifies the fulfillment of the same equation for both thermal and electric conductivities. Distinct points in time, specifically $t=3.75$ s and $t=4.25$ s, showcase the equivalence of the effective radial and tangential thermal (electric) conductivities, thereby elucidating the system’s thermal (electric) sensing capability. In the time intervals 0 s $<t<3.75$ s and 4.25 s $<t<8$ s, the radial thermal (electric) conductivity surpasses its tangential counterpart, establishing the scenario $\kappa _r/\kappa _\theta >1$ ($\sigma _r/\sigma _\theta >1$) and illustrating the thermal (electric) concentrating function. Conversely, during 3.75 s $<t<4.25$ s, the radial thermal (electric) conductivity dips below its tangential equivalent, manifesting as $\kappa _r/\kappa _\theta <1$ ($\sigma _r/\sigma _\theta <1$) and effectively demonstrating the thermal (electric) cloaking function. The alignment between thermal and electric fields within these two-material-based checkerboard structures yields congruent functions that remain adjustable over time. This adaptability results in the creation of spatiotemporal multiphysics metamaterials comprised of just two materials, achieving three distinctive functional combinations simultaneously.

The thermal and electric functions can be toggled based on distinct rules. As depicted in Fig. 6.2e, a checkerboard pattern comprises four isotropic materials characterized by parameters $(\kappa _1,\sigma _1)$, $(\kappa _1,\sigma _2)$, $(\kappa _2,\sigma _1)$, and $(\kappa _2,\sigma _2)$. It’s noteworthy that the electric conductivities for checkerboard configurations, whether assembled from two or four isotropic materials (as shown in Fig. 6.2c, g), exhibit the same variations. This observation leads to the conclusions $\sigma _r^*=\sigma _r$ and $\sigma _{\theta }^*=\sigma _{\theta }$. A notable difference emerges concerning the thermal conductivity when comparing the four-material-based checkerboard to its two-material analog. As the even-numbered layers rotate by an angle $\theta (t)$, the central angle defining the staggered region is represented as $\theta _{s}^*(t)$. The shape parameters for the staggered area in the four-material checkerboard are redefined as

$$\begin{aligned} \theta _{s}^*(t)=\left\{ \begin{array}{cc} \Delta \theta -\frac{\Delta \theta }{\tau }t, &{} 0<t<\tau \\ \frac{\Delta \theta }{\tau }t-\Delta \theta . &{} \tau <t<2\tau \end{array} \right. \end{aligned}$$

(6.22)

In the calculation of the checkerboard structure of two materials, replacing $\theta _{s}(t)$ with $\theta _{s}^*(t)$, the effective thermal conductivity of the checkerboard structure of four materials can be obtained.

Figure 6.2h visually illustrates the temporal progression of effective thermal and electric conductivities within the four-material-based checkerboard structure. This configuration incorporates materials with attributes denoted as $(\kappa _1,\sigma _1)$, $(\kappa _1,\sigma _2)$, $(\kappa _2,\sigma _1)$, and $(\kappa _2,\sigma _2)$. Their respective values are $\kappa _1=20~{{\textrm{Wm}}^{-1}{\textrm{K}}^{-1}}$, $\sigma _1=20~{{\textrm{Sm}}^{-1}}$, $\kappa _2=0.05~{{\textrm{W m}}^{-1}{\textrm{K}}^{-1}}$, and $\sigma _2=0.05~{{\textrm{Sm}}^{-1}}$. The theoretical results presented in Fig. 6.2h, analytical assessments indicate that the four-material-based checkerboard structure supports five unique functional combinations. These functions include: Thermal sensing with electric concentrating at $t=0.25$ s and $t=7.75$ s. Thermal concentrating paired with electric sensing at $t=3.75$ s and $t=4.25$ s. Thermal cloaking with electric concentrating before 0.25 s and after 7.75 s. Thermal concentrating combined with electric cloaking between 3.75 s and 4.25 s. At all other instances, thermal and electric concentrating coexist. In summary, spatiotemporal multiphysics metamaterials comprising four materials have the versatility to manifest these five distinct functional combinations.

3.2 Multi-functional Regulation of Thermal and Electric Field

The dynamics of thermal and electric fields within a rotatable checkerboard structure can be precisely forecasted using the spatiotemporal effective medium theory. For experimental corroboration of the functionalities inherent to spatiotemporal multiphysics metamaterials, finite element simulations are conducted. The initial focus is on a rotatable checkerboard setup made of two isotropic materials. This structure takes the form of a hollow cylinder, segmented into 15 sub-layers. Each of these is further partitioned into 24 fan-shaped unit cells, which alternate between the two materials. A distinguishing feature is that the even-numbered layers of this configuration rotate clockwise as time progresses, as illustrated in Fig. 6.2a. To determine the radius $r_i$ of each layer, the equation $\ln (r_{i+1}/r_i) =\eta \Delta \theta$ is employed, where $\eta =1/\sqrt{10}$, the innermost layer has a diameter of $r_1=4$ cm, and the central angle per unit is given by $\Delta \theta =2 \Pi /24$. This checkerboard construct is set within a square boundary with 45 cm sides, and thermal impedance between units is neglected. Temperature and potential gradients applied at the boundaries generate heat and electric currents throughout. The left boundary holds a temperature and potential of 373 K and 10 mV, respectively, while the right boundary remains at 273 K and 0 mV. Importantly, the overall thermal and electric conductivities of this spatiotemporal metamaterial coincide with theoretical anticipations. With a rotation duration of $2\tau =8$ s, the system undergoes two full rotations. The outcomes of these simulations corroborate the theoretical projections, confirming the functional viability of spatiotemporal multiphysics metamaterials.

Fig. 6.3

Illustrative diagrams and simulation outcomes showcasing a checkerboard structure grounded in two distinct materials. In panels a–c, the structures at time instances $t=0$ s, $t=3.75$ s, and $t=4$ s. Here, I, II, and III denote the central, checkerboard, and background regions, respectively. Correspondingly, panels d–f exhibit simulated thermal field results, with blue lines tracing isotherms. Furthermore, panels g–i present simulation findings of the electric field, with white lines indicating equipotential lines. Notably, panels j and k capture data from the simulation outcomes’ horizontal centerline in d–f and g–i. Particularly, in region III, the three temperature (or potential) lines converge, as depicted. (from Ref. [57], licensed under CC-BY 4.0)

The introduced theory offers a predictive framework for the behaviors of thermal and electric fields at any specific time. Using just two materials, spatiotemporal multiphysics metamaterials can manifest three unique functional combinations, as highlighted in Fig. 6.3. For demonstration, three time snapshots–corresponding to different functional combinations within the checkerboard structure–were chosen for simulation: $t=0~{\textrm{s}}$, $t=3.75~{\textrm{s}}$, and $t=4~{\textrm{s}}$. In the schematic presented in Fig. 6.2d, these instances represent concentration, sensing, and cloaking functions, respectively.

Regardless of the time point, the background’s isotherms and equipotential lines remain consistent, indicating that the background heat and electric currents are undisturbed. At the initial time mark, the even-numbered layers are static, resulting in a checkerboard pattern made of alternating, uniform fan-shaped segments. Simulation outcomes manifest the phenomena of thermal and electric concentration, where the heat and electric currents focus on the central region without perturbing the background. As seen in Fig. 6.3j, k, the central area displays heightened temperature and electric potential gradients relative to the background, confirming the concentration effects of both thermal and electric fields.

Transitioning to $t=3.75~{\textrm{s}}$, the temperature and potential gradients in the central region replicate those of the background, showcasing thermal and electric sensing functionalities. When $t=4~{\textrm{s}}$, the entire checkerboard structure assumes a composition of alternating materials. The simulation outcomes depict the absence of isotherms and equipotential lines within the central region. Temperature and potential gradients in this zone approach zero, confirming the achievement of thermal and electric cloaking. Refer to Fig. 6.3j, k. It’s noteworthy that the isotherms and equipotential lines in region II of the spatiotemporal metamaterial at $t=4$ s exhibit more irregularities compared to those at $t=3.75$ s. This phenomenon can be attributed to the absence of a uniform region in the alternating structure at $t=4$ s. Continuous time adaptation of thermal and electrical functions is achieved by using a rotatable checkerboard structure. Consequently, the simulation verifies the reliability of the theoretical framework, effectively showcasing that spatiotemporal multiphysics metamaterials composed of just two materials have the capacity to achieve three diverse functional combinations.

Fig. 6.4

Schematic diagrams and simulation results of a four-material-based checkerboard structure. Panels a–e showcase the structures at time points $t=0~{\textrm{s}}$, $t=0.25~{\textrm{s}}$, $t=2~{\textrm{s}}$, $t=3.75~{\textrm{s}}$, and $t=4~{\textrm{s}}$. Panels f–j exhibit simulated thermal field results, while panels k–o present simulation outcomes of the electric field. Panels p and q present data derived from the simulation outcomes’ horizontal centerline in f–j and k–o. Notably, within region III, the five temperature (or potential) lines converge, as illustrated. (from Ref. [57], licensed under CC-BY 4.0)

Thermal and electric fields can exhibit varying functionalities. Introducing a unique checkerboard design with four isotropic materials, parameters are aligned with those depicted in Fig. 6.2h. The simulation outcomes at key moments–$0~{\textrm{s}}$, $0.25~{\textrm{s}}$, $2~{\textrm{s}}$, $3.75~{\textrm{s}}$, and $4~{\textrm{s}}$–are showcased in Fig. 6.4. The left boundary conditions for temperature and potential are fixed at 373 K and 10 mV, respectively, while the right boundary maintains 273 K and 0 mV. Significantly, the background remains consistently unperturbed in terms of heat and electric flow.

At the outset, $t=0~{\textrm{s}}$, the heat flow avoids the central region, while electric currents converge, embodying thermal cloaking alongside electric concentration. Conversely, at $t=4~{\textrm{s}}$, electric currents bypass the core as the heat flow concentrates, signifying electric cloaking with thermal concentration. The time frame of $t=0.25~{\textrm{s}}$ demonstrates thermal sensing paired with electric concentration, and $t=3.75~{\textrm{s}}$ showcases thermal concentration coupled with electric sensing. Crucially, the simulated results resonate with theoretical expectations. By $t=2~{\textrm{s}}$, the even-numbered layers have rotated by half of the unit’s central angle, as illustrated in Fig. 6.2h. As a result, radial thermal (and electric) conductivity surpasses its tangential counterpart. Both heat and electric currents center on the core region, representing concurrent thermal and electric concentration. As supported by Fig. 6.2h, the phase from $0.25~{\textrm{s}}$ to $3.75~{\textrm{s}}$ exemplifies concurrent thermal and electric concentration. This demonstrates that spatiotemporal multiphysics metamaterials built from four distinct materials can achieve five functional combinations.

4 Conclusions and Outlook

In this chapter, we have introduced an innovative approach to designing spatially modulated thermoelectric metamaterials, grounded in the principles of transformation theory. We have explored various transformation rules encompassing coupled thermoelectric fields, decoupled thermoelectric fields, and nonlinear thermoelectric fields. Through the strategic application of spatial transformations, we have successfully achieved the fusion of thermal and electric fields into a single unified function.

To amplify the functionality and versatility of these metamaterials, we have put forth the concept of a spatiotemporal multiphysics metamaterial employing a rotatable checkerboard structure. This structure unlocks the capability for simultaneous modulation of thermal and electric fields. By introducing a temporal dimension, specifically encapsulated in the rotation time, we enable a continuous and dynamic control over effective thermal and electric conductivities. This temporal manipulation facilitates seamless transitions between diverse functions of thermal and electric fields, fostering enhanced adaptability and performance.

The spatiotemporal checkerboard structure, whether composed of two or four materials, not only offers three or five distinct function combinations, respectively, but also showcases remarkable potential for advanced applications. As we look forward, the integration of these principles with emerging technologies and materials could open up new avenues for efficient energy management, multifunctional devices, and transformative solutions across multiple industries. The interplay between spatial and temporal dimensions within metamaterials presents a promising frontier for researchers and engineers alike, guiding us towards groundbreaking innovations in the realm of functional materials and device design.