Roadmap on thermodynamics and thermal metamaterials

Abstract

Thermal metamaterials represent a transformative paradigm in modern physics, synergizing thermodynamic principles with metamaterial engineering to master heat flow at will. As next-generation technologies demand multi-scale thermal control, this field urgently requires systematic frameworks to unify its multidisciplinary advances. Curated through a global collaboration involving over 50 specialists across 25 subdisciplines, this review primarily summarizes two decades of advancements, ranging from theoretical breakthroughs to functional implementations. The review reveals groundbreaking innovations in heat manipulation through the exploration of both classical and non-classical transport regimes, topological thermal control mechanisms, and quantum-informed phonon engineering strategies. By bridging physical insights like non-Hermitian thermal dynamics and valleytronic phonon transport with cutting-edge applications, we demonstrate paradigm-shifting capabilities: environment-adaptive thermal cloaks, AI-optimized metamaterials, and nonlinear thermal circuits enabling heat-based computation. Experimental milestones include 3D thermal null media with reconfigurable invisibility and thermal designs breaking classical conductivity limits. This collaborative effort establishes an indispensable roadmap for physicists, highlighting pathways to quantum thermal management, entropy-controlled energy systems, and topological devices. As thermal metamaterials transition from laboratory marvels to technological cornerstones, this work provides the foundational lexicon and design principles for the coming era of intelligent thermal matter.

Graphical abstract

Keywords

thermal metamaterials / thermal cloaking / transformation thermotics / heat conduction control / topological thermotics / thermal rectification / heat transfer

1 Opening remarks

Yuguang Qiu, Jiping Huang^*

　 Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China

Since the birth of human civilization, heat has been the most widely studied and most used energy source, and the source of countless inventions has come from humanity’s thirst for thermal energy regulation. Driven by this grand vision, thermal metamaterials emerged, opening up a new era of unprecedented thermal transport control with their unique, nature-defying thermal properties. Over the past decade, the field of thermal metamaterials has blossomed, with theoretical breakthroughs and engineering applications flourishing, showcasing tremendous potential and value.

Such vibrant development is largely due to continuous communication and collaboration among researchers. As part of this dynamic exchange, we recently held the biennial International Conference on Thermodynamics and Thermal Metamaterials (ThermoMeta2024), which took place in Wuzhen, Zhejiang, from 9 to 12 in May, 2024. The ThermoMeta International Conference series started in 2020 and takes place every two years. The first two events, in 2020 and 2022, were held online because of the COVID-19 pandemic. By the time the 2024 conference happens in person, three successful events have taken place. Over 200 invited speakers from more than 10 different countries have participated in these conferences, and many of them are authors of this review article.

As a Roadmap designed to guide the direction of future research, this article does not attempt to provide an exhaustive review of the entire history of thermal metamaterials. Rather, it aims, as its name suggests, to establish a macro framework that outlines the clear trajectory of thermal metamaterials’ development for scholars both within and outside the field. Within the broader context of thermodynamics and thermal metamaterials, each chapter’s author has selected research topics based on their expertise and interests, with each topic representing a unique branch of the field. These branches can be categorized based on heat transfer mechanisms such as conduction, convection, radiation, and multi-transfer coupling; or according to the types of physical fields, ranging from single thermal systems to multi-physics interactions like thermoelectrics and optoelectronics. Some chapters delve into specific attributes of physical fields, such as nonlinearity, topology, and non-Hermitian concepts; while others focus on fundamental theoretical research in thermodynamics. Although different chapters may discuss similar works, each author brings a distinct perspective to the same body of work. These unique viewpoints reflect various development paths within the field of thermal metamaterials, and when combined, they offer a rich and intertwined picture. This allows us to present both the overall landscape of the field and the distinct evolutionary journeys of its sub-branches — an approach we hope will resonate with readers.

Therefore, the purpose of this Roadmap goes beyond merely summarizing past developments and projecting future trends. It aims to build a bridge between the past and the future, offering valuable insights and experiences for future researchers. We hope this Roadmap will serve as a beacon for the ongoing development of thermal metamaterials, illuminating the way forward and guiding more aspiring scholars to jointly create a brighter future.

In particular, this review article encompasses two major parts: fundamental and applied. The former focuses on thermodynamics, while the latter, as an application of the former, centers on thermal metamaterials. For the applied part, the primary emphasis is on three aspects: physics, materials, and devices. However, it should be noted that different chapters may address the same aspect, while a single chapter may cover multiple aspects. Given this complexity, the article does not organize the chapters based on content categorization. Instead, the chapters are largely arranged according to the submission dates of the final manuscripts. This arrangement has the advantage that each chapter in the main body of the article (Chapters 2–26) is independent of the others, see Fig.1. Readers can therefore select any chapter to begin reading without worrying about whether they have read the preceding chapters. Undoubtedly, this approach is beneficial for readers.

2 Heat conduction control using phononic crystals

Masahiro Nomura^*

　 Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan

2.1 Background

Phononic crystals (PnCs) are artificial periodic structures designed to control the propagation of phonons − the quantum of lattice vibrations responsible for heat conduction in solids or acoustic wave. By engineering the periodicity and geometry of PnCs, researchers aim to manipulate phonon transport and thereby control thermal properties in ways not possible with natural materials. This approach offers new possibilities for thermal management in nanoelectronics, thermoelectric energy conversion, quantum science, and other applications where precise control of heat flow is critical [1, 2].

The concept of PnCs for thermal applications emerged from earlier work on PnCs and acoustic wave control [3–7]. However, thermal phonons present unique challenges due to their short wavelengths (typically a few nanometers at room temperature) and wide frequency spectrum (from gigahertz to terahertz). These characteristics necessitate the development of nanostructures with feature sizes comparable to or smaller than the phonon wavelengths and mean free paths [8]. Early theoretical studies in the late 1990s and early 2000s proposed using superlattices and other nanostructured materials to modify phonon dispersion and reduce thermal conductivity [9]. These works laid the foundation for understanding how periodic structures could influence phonon transport through mechanisms such as zone folding, mini-band formation, and coherent interference effects [10, 11]. Experimental realization of thermal PnCs followed in the 2010s, enabled by advances in nanofabrication techniques or measurement at low temperatures. These developments allowed researchers to create structures with periodicities and feature sizes on the order of tens to hundreds of nanometers, suitable for interacting with thermal phonons [12, 13].

The study of thermal transport in PnCs intersects with several broader areas of research in nanoscale heat transfer. These include ballistic and quasi-ballistic phonon transport in nanostructures, coherent phonon effects and the wave nature of heat, phonon confinement and modification of phonon dispersion, interface and boundary scattering of phonons, and size effects on thermal conductivity [14]. Understanding and exploiting these phenomena in PnCs has been a major focus of research, driving both fundamental scientific insights and potential technological applications.

In this chapter, we will explore PnCs and their role in heat conduction control. We will trace the development of PnCs from their conceptual origins to current research, highlighting key theoretical and experimental advances. This overview will illuminate how PnCs are revolutionizing thermal management and opening new avenues for thermal engineering.

2.2 Past to current development

Research on thermal PnCs has progressed rapidly over the past decade, with key developments in theory, simulation, fabrication, and measurement techniques. Here we highlight some of the major advances.

2.2.1 Theoretical and computational modeling

Researchers have developed various approaches to model phonon transport in PnCs, including the Boltzmann transport equation (BTE), non-equilibrium Green’s function (NEGF) methods, and molecular dynamics (MD) simulations [15]. These tools have provided insights into the fundamental mechanisms of phonon transport in periodic nanostructures, including coherent effects like wave interference and localization.

The BTE approach has been particularly useful for studying the transition between coherent and incoherent phonon transport regimes in PnCs. Simkin and Mahan proposed a model that captures the crossover from wave-like to particle-like behavior as a function of superlattice period length [16]. This work helped explain the observed minimum in thermal conductivity for certain superlattice structures.

NEGF methods have been instrumental in investigating quantum transport in PnCs, especially for structures with dimensions comparable to or smaller than the phonon coherence length. These techniques have revealed phenomena such as phonon localization and the formation of mini-band structures in superlattices.

MD simulations have provided atomic-scale insights into phonon transport in PnCs, allowing researchers to study the interplay between coherent wave effects and incoherent scattering processes. MD studies have been particularly valuable for investigating the impact of interface roughness and defects on thermal transport in nanostructured materials [17, 18].

2.2.2 Fabrication of nanoscale PnCs

Fig.2 shows examples of PnC structures. For 1D PnCs, epitaxial growth techniques allow for the creation of superlattices with atomically sharp interfaces and precise control over layer thicknesses. This has been crucial for studying coherent phonon transport effects in one-dimensional (1D) PnC systems such as GaAs/AlAs [19], and oxide superlattices [20]. Advances in top−down nanofabrication techniques like electron beam lithography have enabled the creation of 1D [21] and 2D [12, 22] PnCs in thin membranes of materials like Si. These structures typically consist of periodic arrays of holes or pillars etched into suspended membranes or deposited structures on membranes [23], allowing for precise control over the phonon scattering landscape. Bottom-up approaches using self-assembly have also been demonstrated. For example, block copolymer lithography has been used to fabricate silicon nanomesh structures with sub-20 nm feature sizes, exhibiting exceptionally low thermal conductivity [13].

2.2.3 Measurement techniques

Experimental methods have been developed to probe thermal transport in PnC structures. Optical pump-probe techniques, such as time-domain thermoreflectance (TDTR) has enabled the measurement of thermal properties. This techniques have been crucial for studying the cross-plane thermal conductivity of superlattices and other layered PnC structures. In-plane thermal conductivity measurement for 2D PnC structures requires free-standing membrane structure to avoid heat leakage to substrates. There are two methods: the electrical 3\omega method [13] and the fully optical \mu-TDTR method [21]. The former allows for highly sensitive measurements, while the latter enables high-throughput thermal conductivity measurements.

Thermal conductance measurements at ultra-low temperatures have allowed researchers to isolate and study coherent phonon transport effects in PnCs, providing direct evidence for the modification of phonon dispersion and the emergence of phononic bandgaps [24].

Brillouin light scattering has been used to directly measure phonon dispersion in PnCs, providing experimental validation of theoretical predictions and insights into the formation of phononic bandgaps [25, 26]. Raman thermometry has also been employed to measure local temperature distributions in PnCs, allowing for the study of size effects and boundary scattering on thermal transport [27].

2.2.4 Coherent thermal transport in PnCs

A major milestone in the field was the experimental observation of coherent phonon transport in superlattices, evidenced by a non-monotonic dependence of thermal conductivity on period length [20]. This work, conducted on epitaxial oxide superlattices, showed a minimum in thermal conductivity at a critical period length, confirming theoretical predictions of a crossover between wave-like and particle-like phonon transport regimes. Subsequent studies have explored coherent thermal transport in 2D PnC systems. These include Si nanomesh and circular hole array structures, where the interplay between coherent and incoherent phonon scattering processes was investigated [12, 24, 28]. Pillar-based PnCs have also been studied, where local resonances can couple with propagating phonon modes to create hybridized states and modify thermal transport [29]. Although experimental realization is not so easy [30, 31], these demonstrations have opened new possibilities for wave-based control of heat transport, analogous to the manipulation of light in PnCs.

2.2.5 Ultralow thermal conductivity

PnCs have enabled the realization of thermal conductivities below the amorphous limit in crystalline materials. This has been achieved through a combination of nanostructuring to limit phonon mean free paths and coherent effects to modify phonon dispersion [13, 32]. Key results in this area include Si nanomesh and PnC structures with extremely low thermal conductivities, which can be attributed to the band-folding effect of phonons and surface scattering.

Researchers have also demonstrated holey Si thin films with thermal conductivities tunable over two orders of magnitude through control of the hole size and spacing. Nanopillar-based PnCs have shown significant reduction in thermal conductivity due to a combination of boundary scattering and coherent effects. These achievements have important implications for thermoelectric materials [34], thermal insulation [33], and thermal management in nanoelectronics. Hu et al. [35] utilized machine learning to design GaAs/AlAs superlattices that minimize cross-plane thermal conductivity. The optimized aperiodic structure effectively suppressed coherent phonon transport, achieving lower thermal conductivity than conventional periodic structures. This approach demonstrates potential for thermal conductivity control in various nanostructured materials, including 2D materials [35].

2.2.6 Thermal phonon engineering in 2D materials

The unique properties of 2D materials have been exploited to create PnCs with novel thermal properties in graphene [36]. Xiao et al. [37] fabricated suspended crystalline MoS₂ membranes by focused ion beam to develop PnCs and observed remarkably reduced thermal conductivity. They also realized MoS₂-based thermal routing nanostructures for thermal isolation and heat guidance in prespecified directions. This approach can be extended to other layered materials like graphene or hBN, opening up innovative strategies for thermal management.

Exploration of topological phonon states in patterned 2D materials has offered the possibility of robust, direction-dependent thermal transport. Studies of strain engineering in 2D material PnCs have shown promise for modulating phonon transport and creating thermal gradients at the nanoscale [38]. The atomically thin nature of these materials, combined with their high intrinsic thermal conductivity and mechanical strength, makes them particularly promising for next-generation thermal management and energy conversion applications.

Mathematical concepts have inspired novel approaches to engineering thermal transport in 2D materials. For instance, the Golomb ruler sequence has been utilized to design graphene/h-BN heterostructures that efficiently suppress coherent phonon transport and thermal conductivity more than periodic structures [39]. Similarly, isotope interfaces in graphene based on the Golomb ruler pattern have demonstrated strong phonon scattering and confinement effects, leading to reduced thermal conductivity at cryogenic temperatures [40].

2.2.7 Inverse design of PnC

The design of phononic crystals has traditionally relied on intuitive concepts and simple geometries, limiting their potential for tailored wave manipulation. Recent advances in inverse design methodologies have revolutionized this field, enabling the creation of phononic structures with unprecedented functionalities [41]. Inverse design techniques, such as topology optimization and machine learning, can be applied to phononic meta-structured materials design. These methods explore a significantly larger parameter space compared to conventional approaches, often leading to unintuitive unit cell shapes with superior performance. An example of this approach is demonstrated by a genetic algorithm to design a 2D PnC with highly anisotropic phonon dispersion [26]. This high anisotropy was achieved through genetically optimized unit cells with shapes exceeding conventional intuition. This work highlights the potential of inverse design in creating phononic structures with customized functionalities, such as efficient heat transport, miniaturized acoustic lenses, and negative refraction devices. Inverse design methodologies open up new possibilities in the field of PnCs, enabling the exploration of phonon manipulation and high acoustic Q-factor achievements for quantum applications [42].

2.3 Future outlook

Several promising directions are emerging for thermal PnCs research. While most work to date has focused on 1D and 2D structures, 3D PnCs offer more degrees of freedom for phonon engineering. Advances in additive manufacturing and self-assembly may enable the realization of complex 3D architectures optimized for specific thermal functions [43, 44]. New PnC design methods, including mathematically-inspired approach, machine learning approach, quasicrystalline structuring, biomimetics, will create more fuctional and high-performance thermal metamaterials with exceptional thermal properties. The integration of PnCs into 3D electronic and optoelectronic devices including quantum computers for improved thermal management is also an exciting prospect.

The development of PnCs with tunable or switchable properties could enable adaptive thermal management. This might be achieved through mechanisms such as electrostatic gating to modulate phonon transport in semiconductor PnCs, mechanical deformation of flexible or stretchable PnCs to alter their thermal properties, and integration of phase change materials to create thermally reconfigurable PnC structures [45]. Optically controlled PnCs using materials with strong phonon-photon coupling and magnetically tunable PnCs incorporating magnetostrictive or magnetocaloric materials are also being explored. These active PnCs could form the basis for thermal switches, regulators, and logic devices operating on principles analogous to their electronic counterparts.

Exploring nonlinear phonon interactions and topological states in PnCs may reveal new phenomena and functionalities. Researchers are investigating unidirectional heat transport exploiting topological edge states in PnCs, phonon/thermal diodes and transistors based on nonlinear interactions in asymmetric phononic structures, and soliton-like heat transport in engineered nonlinear PnC waveguides. The emerging field of valley phononics, which leverages valley degrees of freedom for heat manipulation, and the development of phonon lasers and amplifiers utilizing stimulated phonon emission in PnCs are also areas of active research [46]. These novel transport regimes could enable unprecedented control over heat flow at the nanoscale, with potential applications in thermal logic and computing.

Combining PnCs with other systems like photonic crystals or magnonic crystals could enable novel multifunctional devices. This includes thermoelectric devices with co-designed electronic and phononic crystals, optomechanical crystals for controlling both light and heat at the nanoscale, and hybrid phoxonic (photon−phonon) circuits for information processing and sensing applications. Researchers are also exploring magnon−phonon coupling in PnCs for spin caloritronics and thermal spintronics, as well as polariton-based thermal devices exploiting strong coupling between photons, phonons, magnons, and electronic excitations. These integrated systems could pave the way for new classes of multifunctional devices that leverage the interplay between different types of quasiparticles for enhanced performance or novel functionalities.

2.4 Summary

Phononic crystals have emerged as a powerful platform for controlling thermal transport at the nano/micro scale. Significant progress has been made in understanding and manipulating phonon propagation coherently and/or incoherently through periodic nanostructures, enabling unprecedented control over thermal properties. The field will expand to the areas such as 3D and dynamic PnCs, nonlinear phenomena, multiple quanta, and integration with other functional systems. As our ability to engineer phonon transport continues to advance, thermal PnCs are likely to play an increasingly important role in addressing thermal management challenges in nanoelectronics, energy conversion, and other critical technologies.

3 Nanophononic metamaterials: Advances in thermal and phonon transport

Zhongwei Zhang¹, Shuang Lu¹, Sebastian Volz^1,2, Jie Chen^1,*

　 ¹Center for Phononics and Thermal Energy Science, China-EU Joint Lab for Nanophononics, MOE Key Laboratory of Advanced Micro-structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

　 ²Laboratory for Integrated Micro and Mechatronic Systems, CNRS-IIS UMI 2820, The University of Tokyo, Tokyo 153-8505, Japan

3.1 Background

Achieving efficient and precise regulation of thermal and phonon transport has important applications in the fields of thermal management in micro/nanoelectronics [47, 48], thermoelectric energy conversion [49, 50], and information communications [51, 52]. Inspired by acoustic, electromagnetic, and elastic wave metamaterials, nanophononic metamaterials (NPMs) have been developed to offer innovative and enhanced mechanisms for controlling thermal and phonon transport, surpassing the capabilities of traditional materials [53–56].

These NPM systems, whether periodic or random, can manipulate phonon propagation through both particle-like and wave-like mechanisms [57–63]. While phonons are traditionally treated as quasi-particles in conventional thermal transport scenarios, their wave-like behaviors become increasingly relevant in nanostructured materials or small semiconductor devices [62, 64, 65], when the characteristic dimensions fall below the phonon mean free path or coherence length. This dual nature of phonons necessitates that both particle-like and wave-like properties should be considered in the design of NPM, distinct from other types of design [66–71].

To date, various NPM systems have been proposed [68, 69, 72–79], which can be categorized into two main groups based on their objectives. The first category focuses on regulating thermal transport by controlling phonon propagation, including reducing thermal conductivity through resonance mechanisms and enhancing interfacial thermal resistance via interfacial nanostructuring [68, 69, 72, 73]. The second category targets manipulating phonons at the specific frequency, with applications such as designing novel phononic devices [74, 75], enabling information transmission [76, 77], and realizing solid-state quantum chips and quantum computing [78, 79]. Numerous theoretical and experimental studies have extensively explored the influence of structural randomness, atomic disorder, and temperature effects on thermal and phononic properties [80–84]. In this chapter, we will review the development of NPM from both theoretical and experimental perspectives, and provide an outlook for the future studies.

3.2 Nanophononic metamaterials for manipulating thermal transport

In this section, we review the advancements in regulating thermal transport through NPM. Over the past few decades, the development of NPM for this objective has progressed rapidly. The modulation of thermal transport in these artificial structures primarily relies on Bragg scattering and resonance mechanisms. Nanomeshes utilizing Bragg scattering have been both theoretically and experimentally validated for their effectiveness in regulating thermal conductivity [85–89] and have been extensively discussed in various reviews [90–92]. Consequently, this section will focus on the resonant type NPM. Resonant type NPM, especially the pillar-shaped structures and host−guest system, have a significant impact on thermal conductivity, attracting widespread attention [67, 92–98]. Moreover, the design of interface NPM to either enhance or reduce interfacial thermal resistance has also flourished, highlighting their immense potential in thermal management applications.

3.2.1 Pillar-based nanophononic metamaterials

The concept of pillar-based NPM for reducing thermal conductivity was introduced by Davis and Hussein in 2014 [67, 93], inspiring further studies that deepened the understanding of the underlying physics and performance [67, 68, 99–104]. As shown in Fig.3, the unit cells for silicon, uniform membrane, single-pillared, and double-pillared NPM are presented. The dimensions of the membrane and nanopillars are characterized by the variables a{A}_x, a{A}_y, d and b,b,h_T, respectively. The nanopillars act as resonators that are coupled with heat-carrying phonons, facilitating resonance hybridization between wavenumber-independent resonant modes (i.e., vibrons) and wavenumber-dependent propagating modes (i.e., phonons).

The reduction in thermal conductivity induced by the nanopillars can be attributed to three atomic-scale mechanisms. Firstly, local resonance flattens the dispersion curves, resulting in the reduced group velocity and enhanced energy localization. Secondly, increased phonon−phonon scattering shortens phonon lifetimes, further reducing mean free paths and thermal conductivity. Thirdly, the top and bottom surfaces of the unit cell may not be perfectly smooth, indicating the enhanced boundary scattering at these surfaces, including those of the nanopillars.

The first effect is theoretically demonstrated through lattice dynamics calculations on the unit cell of the NPM. The dispersion and group velocity diagrams reveal a significant reduction in phonon group velocities, particularly at points where resonance hybridization occurs and in the surrounding coupling regions. Fig.4(b) further illustrates a specific mode in the band structure, comparing its shape with and without resonance hybridization. This comparison underscores the importance of resonance hybridization in localizing vibrational energy within the nanopillars, which plays a crucial role in reducing thermal conductivity.

Additionally, the spectral energy density (SED) method, which accounts for anharmonic effects of phonons at finite temperatures, was employed by Honarvar and Hussein [99] to demonstrate the local resonance effect, as shown in Fig.5. Notably, the phonon dispersion of SED from molecular dynamics (MD) simulations provides clear evidence of vibrons and phonon-vibron coupling. For the components of the membrane base, the SED spectrum in Fig.5(b) reveals that the nanopillars can significantly affect the propagation of travelling phonon waves within the membrane. Furthermore, the broadening of SED spectrum for pillar-based NPM [Fig.5(b)] is notably large compared to that of a uniform membrane [Fig.5(a)], particularly near the frequency region of phonon hybridization, indicating that surface pillars result in intensive phonon scattering.

Since the initial proposal of pillar-based metamaterials for phonons, numerous studies have explored a variety of configurations [68, 103, 105–110]. For instance, the dimensions of nanopillars have been extensively studied, showing a pronounced impact on the thermal conductivity [68, 103, 111, 112]. Liu et al. [107] studied pillar-based metamaterials in the form of branched nanoribbon materials composed of molybdenum disulfide ({\text{MoS}}_2). They found that the side branches not only significantly block phonon transport but also lead to a notable increase in the Seebeck coefficient in both armchair and zigzag MoS₂. Ma et al. [108] studied local resonant effects in Si nanowire cages (SiNWCs) and reported an ultralow thermal conductivity of 0.173\text{W}\cdot {\text{m}}^{-1}\cdot {\text{K}}^{-1}—an order of magnitude lower than that of Si nanowires. This reduction is due to phonon localization from local resonance and hybridization at the junction.

On the other hand, significant debate still persists regarding the influence of resonant pillars on thermal transport and the underlying mechanisms. For instance, some experimental observations [113–117] report that the reduction of thermal conductivity is considerably lower than predictions by simulations and theoretical models. Certain studies attribute the observed reduction of thermal conductivity to diffuse surface scattering of phonons induced by the pillars [118, 119], rather than to resonances within the nanopillars or other coherent effects. Furthermore, some experiments [115, 119, 120] reveal that resonant pillar modes occur only at frequencies around several gigahertz (GHz), which is much lower than the terahertz (THz) range spectrum of thermal phonons. These findings contrast sharply with earlier theoretical predictions [67, 93]. Therefore, pillar-based NPM systems warrant further investigation in the future, with particular emphasis on more systematic experimental validation.

3.2.2 Host-guest system

Clathrate systems raise fundamental crystallographic questions regarding the mechanisms of host−guest interactions and their effects on the thermal and electronic properties of the material, attracting significant interest in the pursuit of efficient thermoelectric materials [121–130]. The structure of the host-guest system and its corresponding phonon dispersion are illustrated in Fig.5. Typically, the host framework and guest atoms are coupled through weak interactions. Fig.6 shows that the phonon dispersion can be separated into the contributions of acoustic motion of the host structure and the Einstein-like energy of the guest atoms. When these two modes combine through a weak interaction, an avoided-crossing phenomenon occurs in the dispersion [126, 131, 132].

Among various host−guest systems, clathrates have received much attention as potential high-performance thermoelectric materials, owing to their favorable electronic properties and low thermal conductivity [131, 133–137]. The guest atoms in clathrates can rattle either on-center or off-center within host cages. Experiments further confirm that the rattling motion of guest atoms significantly impacts the thermal conductivity of these materials [73, 138–144].

For example, Sales et al. [131] reported glass-like behavior in {\text{Eu}}_8{\text{Ga}}_{16}{\text{Ge}}_{30} and {\text{Sr}}_8{\text{Ga}}_{16}{\text{Ge}}_{30}, while the isostructural n-type {\text{Ba}}_8{\text{Ga}}_{16}{\text{Ge}}_{30} exhibited “normal” crystal-like thermal conductivity with a low-temperature peak. They observed substantial off-center rattling for Sr and Eu in {\text{Eu}}_8{\text{Ga}}_{16}{\text{Ge}}_{30} and {\text{Sr}}_8{\text{Ga}}_{16}{\text{Ge}}_{30}, whereas Ba vibrates around a centered position in {\text{Ba}}_8{\text{Ga}}_{16}{\text{Ge}}_{30}. The variations in thermal conductivity among these systems arise from the structural arrangement of guest atoms. Resonant phonon scattering and tunneling states contribute to the glass-like behavior in Sr and Eu samples, while Ba’s centered position reduces tunneling effects and weakens coupling to framework modes.

In most clathrate systems, the motion of guest atoms is quite complex, with some clathrate systems potentially containing both off-center and on-center guest atoms simultaneously. For instance, Baumbach et al. [142] confirmed that \text{Eu}(1) is located at the center position, while \text{Eu}(2) is situated at the off-center position. Additionally, it has been suggested that the coupling between guest atoms and the framework varies, depending on the type of charge carrier and the type of the guest atoms. For example, Tang et al. [145, 146] indicated that the composition of the tetrakaidecahedron is significantly influenced by both the type of guest atom and the charge carrier type by X-ray photoelectron spectroscopy analysis. This complexity may further enhance phonon scattering or localization of phonons at various frequencies, consequently reducing the material’s thermal conductivity.

The “rattling” phenomenon has also been widely associated with other similar frameworks, such as perovskites and schwarzites [127–130, 147–152]. For instance, using ab initio MD simulations, Hata et al. [147] demonstrated that in MAPbI₃, the coupled motions of MA⁺ cations scatter heat-carrying phonons, thereby reducing thermal conductivity. In addition, Zhang et al. [153] proposed a novel host-guest system based on schwarzites and studied the relationship between thermal conductivity and the rattling motion of guest atoms in schwarzites. More importantly, they found that the maximum reduction in thermal conductivity can be achieved by matching the frequency of hybridized modes to that of the dominant phonons in thermal conductivity spectrum as shown in Fig.5. This provides theoretical guidance for regulating material’s thermal conductivity via phonon resonance.

3.2.3 Interfacial nanophononic metamaterials

Interfacial thermal transport has gained significant attention because interfaces are becoming the most crucial barrier for heat dissipation in advanced micro/nanoelectronic devices [47, 154, 155]. Efficient heat dissipation is essential for preventing overheating and ensuring the reliability and performance of these devices [76, 156–161]. Interfacial heat transport can be influenced either by interfacial bonding (bonding effect) or vibrational mismatch (bridging effect). Designing NPM that target these aspects has been demonstrated as an effective strategy for regulating thermal transport at interfaces.

Prior theoretical studies have demonstrated the existence of an optimal interfacial coupling strength for maximizing interfacial thermal conductance (ITC). Using a 1D atomic chain model, Zhang et al. [162] found that ITC initially increases with interfacial interaction and then decreases, indicating the presence of optimal coupling as shown in Fig.8(a). Subsequent works by Lu et al. [163] and Xu et al. [164], which considered nonlinear interactions in the 1D atomic chain model, yielded similar results, demonstrating a general trend of the non-monotonic dependence of ITC on interfacial bonding strength.

In realistic materials, self-assembled monolayers (SAMs) can be used to effectively control bonding strength by facilitating the covalent linking between two materials, thereby enhancing bonding at interfaces that are normally held together by weak van der Waals interactions [165–174]. For instance, Losego et al. [165] investigated the interfacial thermal conductance (ITC) of a quartz/gold interface using thiol- and methyl-functionalized SAM. The ITC of the quartz/gold interface with the thiol-functionalized SAM is 68 \text{MW}\cdot {\text{m}}^{-2}\cdot {\text{K}}^{-1}, nearly twice that of the methyl-functionalized SAM, which is around 36\text{MW}\cdot {\text{m}}^{-2}\cdot {\text{K}}^{-1}. This highlights the significant increase in ITC achieved by enhancing bonding through SAM interlayers.

On the other hand, introducing an interlayer to appropriately mediate vibrational mismatch is an effective approach for achieving the bridging effect. For instance, English et al. [175], and Liang and Tsai [176] demonstrated that an interlayer can improve ITC by decreasing vibrational mismatch. Besides, Ma et al. [177] designed graded interlayers, with each layer composed of different materials. They found that these graded interfaces significantly enhance interfacial thermal conductance and pointed out that exponentially mass-graded interlayers are more effective than linearly mass-graded ones (see Fig.8(b)). This improvement is attributed to better vibrational matching between interfacial layers (see Fig.8(c)).

In practical applications, Chang et al. [178] and Liu et al. [179] introduced Ti and Cr interlayers to enhance the interfacial thermal conductance of the copper/diamond interface by reducing vibrational mismatch and strengthening interfacial bonding. Similarly, Hung et al. [159] employed a SAM interlayer to improve the interfacial thermal conductance, primarily by mitigating vibrational mismatch. Sun et al. [180] observed a seven-fold enhancement in ITC for Au/polyethylene interface through thermoreflectance measurements.

Thermal transport at solid−liquid interfaces is influenced by a complex set of factors compared to solid−solid interfaces. Liquid surface wettability and molecular ordering impact both bonding and bridging effects, thereby affecting interfacial thermal conductance [181–184]. For example, Alexeev et al. [183] found that the Kapitza resistance at the graphene−water interface is inversely proportional to the first density peak of the water layer near graphene, reflecting improved acoustic phonon matching.

Although the bonding effect and bridging effect both can improve the interfical thermal transport, they typically impose mutual limitations on each other. For instance, Xu et al. [185] found that controlling vibrational mismatch also impacts interfacial bonding, leading to a lower-than-expected enhancement of interfacial thermal conductance. This trade-off limits the practical application of interfacial metamaterials. This is because the introduction of interlayers not only modifies the vibrational mismatch but also affects the interfacial bonding due to the distinct interatomic bonds between different materials.

3.3 Nanophonic metamaterials designed for manipulating phonon transport

In addition to applications in thermal management, NPM systems are increasingly recognized for their potential applications in various fields such as crystal surface damage detection [74], quantum communication [186], and quantum chips [187]. The wave nature of phonons offers significant opportunities for designing advanced phononic devices, focusing on specific phonon modes. In this section, we will introduce the control of specific phonon modes in NPM systems and discuss the development of novel phononic devices.

3.3.1 Phonon coherence mechanism

The wave-like nature of phonons enables control over their transmission through constructive and destructive interference. When two coherent phonons are out of phase, they interfere destructively, minimizing reflection and maximizing transmission. Conversely, when they are in phase, constructive interference occurs, leading to reduced transmission. This technique can allow for the complete transmission or reflection of phonons within specific frequency ranges [74, 188–193].

Kosevich et al. [192] theoretically demonstrated that phonon wave interference between different paths can be achieved by designing phononic metamirrors and meta-absorbers. Similarly, Hu et al. [189] realized destructive interference between two different phonon wave paths using a multilayer array of germanium nanoparticles embedded in a crystalline silicon matrix, as illustrated in Fig.9(a). Jiang et al. [193] investigated directly the coherent interference effect in a graphene superlattice structure via wave-packet simulations. They found that the constructive and destructive interference of reflected phonons results in complete reflection and complete transmission in the transmission coefficient, causing the periodic oscillation in the transmission function as the superlattice period length varies.

3.3.2 Nanoscale phononic devices

NPM systems can also be employed to design devices with specific functionalities by targeting particular phonon modes. It has revealed a variety of novel physical phenomena, showing great potential for future applications in microscale and nanoscale devices.

NPM systems excel in absorbing, storing, and emitting phonon energy through innovative designs [74–76, 158, 188, 194–198]. For instance, Zhang et al. [75] introduced a tunable phonon nanocapacitor using a carbon schwarzite host-guest system, capable of storing and emitting monochromatic coherent phonons across GHz to THz frequencies [see Fig.9(b)]. They also demonstrated a phononic rectification system with an efficiency of 134%, surpassing most nanostructures [158]. Yoon et al. [196] also achieved high-Q THz phononic cavities with Van der Waals heterostructures.

Furthermore, some phononic devices based on NPM systems can also be used to control the propagation path and direction of phonons. For example, Cheney et al. [194] and Patel et al. [198] developed phonon transport wires in silicon chips for coherent transport over millimeter distances [see Fig.9(c)]. Lu et al. [76] demonstrated phonon anomalous transmission via a phononic metagrating [see Fig.9(d)]. An intriguing nonlocal effect has emerged due to the inherently strong covalent bonds, accompanied by the occurrence of mode conversion, highlighting the unique characteristics of the phonon system.

In addition, phononic nanodevices hold immense potential in quantum communication and quantum devices, enabling effective transmission and processing of quantum information and facilitating advancements in quantum computing and quantum networks. For instance, Zivari et al. [187] realized quantum entanglement between two traveling phonons using an optomechanical cavity to generate a mechanical quantum state. O’Brien et al. [186] demonstrated that phononic nanodevices can be employed to realize phononic qubits, which serve as fundamental units in quantum computation and quantum information processing.

NPM systems in the THz frequency range have given rise to numerous novel physical phenomena and potential applications. However, due to the practical challenges associated with manipulating phonons in this range, this area of research remains primarily theoretical and simulation-based. Further exploration and experimental validation are needed to advance this field.

3.4 Summary and outlook

In this chapter, we have reviewed management of thermal and phonon transport using NPM, emphasizing resonance-based thermal control in pillar designs, optimized thermal resistance in interfacial materials, and phonon mode control. Key advancements in phononic devices and transport mechanisms are highlighted. However, numerous practical challenges persist in the design and fabrication of NPM, indicating a need for further study. The major challenges, opportunities, and open questions for future research include:

(i) Most NPM systems leverage the wave-like nature of phonons, but the particle-like behavior also significantly influences phonon transport and should not be overlooked. Therefore, further development of theoretical models is essential to capture the dual effects of both coherent phonons and phonon scattering in NPM, as well as to explore materials that can concurrently tune these phonon behaviors.

(ii) Most studies of pillar-based metamaterials focus on periodic lattice arrangements, but practical experimental preparations often result in disordered pillar distributions, which lack sufficient theoretical and simulation exploration. These disordered systems may exhibit richer physical phenomena, offering opportunities for breakthroughs in applied physics and related fields, thus warranting greater attention in future research.

(iii) In acoustic and optical systems, nonlocal metamaterials have seen rapid development, achieving broad bandwidth control [199–202]. Introducing nonlocal effects into NPM to regulate phonon thermal transport can be an important direction for future advancements in NPM.

(iv) Deep learning is an artificial intelligence approach that explores new frontiers in data recognition and processing [106, 203–205]. It is rapidly advancing and permeating various disciplines. While the use of machine learning methods to predict material properties from their structures is already well-established, the inverse design of NPM still requires further exploration, particularly in predicting material structures based on specific customized requirements.

4 Characteristics and mechanisms of heat flux regulation based on nanophononic metastructures

Jian Zhang^1,2,3, Haochun Zhang^1,*

　 ¹School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

　 ²School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China

　 ³Yangtze Delta Region Academy of Beijing Institute of Technology (Jiaxing), Jiaxing 314019, China

4.1 Background

Macroscale heat transfer is characterized by empirical and phenomenological representations of Fourier’s laws, however, nanoscale heat transfer exhibits distinct differences. Notably, Fourier’s law is inadequate at this scale, as there exists a scaling relationship between thermal conductivity and system size, alongside phenomena such as thermal rectification [207, 208]. At the nanoscale, phonons serve as the primary energy carriers in semiconductor materials. Phonons, which are massless and chargeless, exhibit wave-particle duality, complicating the regulation of heat transfer processes at this scale. Research conducted by Li et al. [209] has demonstrated that phonon heat transfer in crystalline structures can facilitate information processing and transmission, leading to the development of concepts such as thermal transistors, thermal diodes, and thermal logic gates. These advancements provide a theoretical framework for utilizing phonons in information processing and mark a significant progression in the regulation of nanoscale heat transfer. As electronic devices become increasingly integrated, the heat flux density in electronic devices has increased sharply, seriously affecting their service life. Effective thermal management is essential for maintaining the stable operation of electronic components. One prevalent strategy in contemporary thermal design is the thermal isolation of heat sources, which involves directing heat flux in a regulated manner.

Heat flux regulation pertains to the manipulation of heat flow direction to either circumvent or direct heat to specific targets. Common structures employed for heat flux regulation include thermal rectification [4], thermal cloaks [211–214], and thermal concentrators [215, 216]. Thermal rectification refers to the phenomenon where heat flux differs in two directions under an identical temperature gradient, achieved through the design of anisotropic thermal conductivity structures. This phenomenon holds significant potential for applications in thermal switches and transistors. Conversely, thermal cloaks involve the implementation of low thermal conductivity regions that allow heat to bypass a designated region, driven by temperature differentials, thereby preserving the temperature of that region. This concept is particularly relevant for the thermal isolation of sensitive components, although its design poses challenges in effectively creating low thermal conductivity regions. Thermal concentrators complement thermal cloaks by directing heat flux towards a specific region, thereby maintaining a high heat flux in that region, which can significantly enhance the performance of thermoelectric generators.

The following sections will briefly summarize the development of this field from both theoretical and application perspectives, and provide an outlook on future directions.

4.2 Past to current development

4.2.1 Theory

The fundamental principle underlying the regulation of nanoscale heat flux is the manipulation of the thermal conductivity of materials. Commonly employed techniques for altering the thermal conductivity include the amorphization of nanofilms [217–219], as well as the strategic arrangement of nanoholes [220, 221] and nanopillars [222–233] within these films. This chapter subsequently reviews the mechanisms of regulating thermal conductivity by the above three methods.

In crystalline structures, thermal energy is primarily transported by phonons [234]. However, due to the necessity of periodicity for the definition of phonons, thermal transport mechanisms differ significantly in structurally disordered amorphous materials [235]. Allen et al. [236] conducted lattice dynamics calculations and found that heat carriers can be divided into propagons, diffusons, and locons. Locons are spatially localized high-frequency modes whose contribution to heat conduction is negligible. He et al. [217] used non-equilibrium molecular dynamics (MD) simulations and found that propagons contribute approximately half of the thermal conductivity in amorphous silicon. For amorphous carbon, Lv et al. [218] proposed that if 6.5 THz is the transition frequency from propagator to diffuser, the propagons can contribute 13% to the thermal conductivity. Zhou et al. [219] used Boltzmann transport theory to calculate the thermal conductivity contributed by propagons. Taking 2.47 THz as the transition frequency from propagator to diffuser, the thermal conductivity contributed by propagons in cement-disordered glue exceeded 30%. Therefore, the amorphization of crystalline materials can significantly enhance the strong localization of heat carriers, thereby reducing the thermal conductivity of the material.

The phonon mean free path (MFP) is the average distance that a phonon travels between two scatterings, which can occur from impurities, defects, electrons, and boundaries [237]. Phonon MFP and group velocity are important factors affecting thermal conductivity, which can be regulated using phononic crystals with submicron feature sizes [220, 221]. The reduction in phononic MFP is attributed to imperfections within the periodic structure and the matrix of the phononic crystal. The second-order periodicity of phononic crystals changes the phonon dispersion relationship, leading to a reduction in the phonon group velocity, changing the density of states (DOS), and changing the phonon band gap under certain conditions [238]. Yang et al. [239] studied ballistic heat transport in porous metamaterials and found that an ordered structure perpendicular to the transport direction has a stronger impact on the effective thermal conductivity than an ordered structure parallel to the transport direction. Furthermore, effective thermal conductivity diminishes significantly as cluster size increases. Porous films are used to design thermal rectifiers since they can provide both temperature and spatial dependence of thermal conductivity. It is reported that the thermal rectification efficiency of perforated graphene is 26% at room temperature [210]. Generally, the introduction of nanoholes into bulk materials will cause a strong phonon interference effect, resulting in a significant reduction in thermal conductivity [34]. This phenomenon is strongly enhanced when nanoholes become disordered [241].

Hussein et al. [242] introduced the concept of local resonance metamaterials into thermal science, which is called nanophononic metamaterials (NPMs). These materials can significantly reduce thermal conductivity through local atomic vibrations, and their potential physical properties have been further investigated [243]. Typical nanophononic metamaterials consist of a crystal film with periodically arranged nanopillars on its surface. The thermal conductivity of the base film is reduced by the local resonance effect of the base film and the nanopillars. The primary mechanisms involved include a reduction in phonon group velocity and phonon lifetime, as well as the localization of atoms in the nanopillars at the resonant frequency [244]. In addition, the DOS distribution of the vibrators in the nanopillars can be adjusted to be highly consistent with the density of states distribution of the phonons in the base film, which further enhances the local resonance effect. Honarvar et al. [222] studied a 9.78 nm thick silicon film through equilibrium molecular dynamics (MD) simulation. Its 586.5 nm high nanopillars covered 79% of the surface region on both sides and reduced the room temperature thermal conductivity by 130 times compared with the corresponding pristine film. Xiong et al. [223] studied silicon NPMs in the form of rods/wires with branched substructures and explored the effects of temperature and alloying. As the temperature increases, the degree of thermal conductivity reduction decreases. This is because the intensity of anharmonic phonon−phonon scattering increases, which reduces the effect of local resonance. And alloying will enhance the local resonance effect, further reducing thermal conductivity. In addition, researchers have studied the localization effect in various types of metamaterials, such as three-dimensional silicon nanowire cages [224], amorphous nanopillars [225], surface oxide films [226], branched nanoribbons [227], carbon nanotubes [228], and graphene ribbons [229–231]. Experimental research is still in its early stages, Spann et al. [232] prepared gallium nitride nanopillars on 200 nm silicon films by molecular beam epitaxy, covering about 55% of the surface, and reducing thermal conductivity by nearly 25%. Maire et al. [233] used electron beam lithography and ion etching to prepare nanowalled silicon nanowires and found that larger nanowalls and smaller baseline thicknesses were more helpful in reducing thermal conductivity.

4.2.2 Application

Based on the above structure, researchers have designed a variety of heat flux regulation structures, including nano-thermal cloaks and heat flux concentrators. The schematic diagram of the heat flux regulation structure based on nanophononic metamaterials is shown in Fig.10.

Performance and optimization of nano-thermal cloaks based on different structures. Cao et al. [211] employed a partial chemical functionalization method to design a thermal cloak based on graphene for the first time, investigating the effects of hydrogen, methyl, and hydroxyl groups on cloaking performance. This successfully opened the door to the design of nano-thermal cloaks, leading the field of nanoscale heat flux regulation and thermal protection of microelectronic devices to become a cutting-edge research direction. Liu et al. [212] used the “in-situ annealing” technology to amorphize part of crystalline silicon to build a nano-thermal cloak. By calculating the ratio of thermal cloaking (RTC), it was found that the RTC was significantly higher than that of the chemically functionalized graphene nano-thermal cloak. In addition, the RTC is proportional to the width of the amorphous ring. By calculating the phonon density of states (PDOS) and mode participation rate (MPR), it was found that the amorphous region produces strong phonon localization, which is the main reason for the cloaking phenomenon. The method of designing nano-thermal cloaks by in-situ annealing the atomic lattice structure of the material is more practical and efficient, potentially paving the way for advanced nano-thermal functionalities and thermal management in nanophotonics and nanoelectronics. Zhang et al. [213] investigated the engineering applications of amorphous nano-thermal cloaks, specifically their performance under triangular wave dynamic temperature boundaries. They found that the amplitude and period of temperature fluctuations significantly influence cloaking performance, smaller amplitudes and longer periods yield better results. For instance, when the width of the amorphous ring is 2 nm, with a temperature change amplitude of 400 K and a period of 60 ps, the RTC can reach 5.16. Additionally, a response temperature evaluation index was proposed to evaluate the disturbance of the cloak structure on the background temperature, indicating that a smaller width of the amorphous ring results in less disturbance to the background temperature.

Zhang et al. [214] developed a nano-thermal cloak utilizing nanoholes. They investigated the effects of the number, diameter, and arrangement of nanoholes on cloaking performance. The RTC exhibited a positive correlation with the number of nanoholes. When both the number and diameter of the nanoholes were fixed, the optimal arrangement was found to be circular. With 24 nanoholes arranged in a circular configuration, the RTC reached 3.6. Furthermore, the response surface method was employed to analyze the impact of various parameters on cloaking performance, resulting in fitting equations for the different influencing factors. At the same time, the interaction between each two different parameters was studied to obtain the optimal parameter selection range. By calculating PDOS and MPR, it was found that the main reason for thermal cloaking was phonon localization. The performance optimization of the cloak structure mainly includes two aspects: one is to improve the stability of the structure, and the other is to improve the thermal cloaking performance. However, these two aspects are a contradiction that is difficult to achieve at the same time. Zhang et al. [245] optimized the performance of the thermal cloak by fixing the region of the nanoholes, reducing the diameter of the nanoholes, and increasing the number of nanoholes. It was found that when the region of the nanohole was fixed, the more nanoholes there were, the larger the sum of the circumferences of the nanoholes. At this time, when the phonons generated by atomic vibrations in the nanohole region are roughly the same, the obstacle to phonon propagation increases, and the cloaking performance is stronger. To improve the stability of the structure, Zhang et al. [246] changed the nanoholes into nanogrooves and built a nano-thermal cloak. When the groove depth is 7 Å, the RTC can reach 12.55. The performance of the nano-thermal cloak is optimized by nanoholes and amorphous structures. One method is to amorphize the nanohole region [247], and the RTC reaches 5.51. The other method is to construct a nanohole and amorphous double-layer structure. The cloaking performance increases with the increase of the width of the amorphous region and the number of nanoholes. When the width of the amorphous ring is 2 nm and the number of nanoholes is 24, the RTC can reach 4.79. The frequency dependence of interfacial heat conduction was studied by calculating the spectral decomposition of the heat flow. Compared with the crystal−crystal interface, the peak of the crystal-amorphous interface is the smallest [248]. In addition, Zhang et al. [249] designed a nano-thermal cloak using a cubic porous structure, with an RTC up to 8. By calculating the PDOS and MPR, it was found that the low-frequency PDOS of the porous structure increased, while the high-frequency PDOS decreased, and the peaks all moved to the left, resulting in strong phonon localization. By fixing the total void ratio of the functional region and changing the void ratio of the minimum structural unit, three types of nano-thermal cloaks were constructed, and the pristine film was selected as a reference to study its cloaking performance. When the total void ratio of the porous structure region is the same, the smaller the void ratio of the minimum structural unit, the better the cloaking effect. When the void ratio of the minimum structural unit is 50%, the ratio of thermal cloaking can reach 9.32 [250].

Zhang et al. [251] addressed the issue of interfacial thermal resistance in local heat flux regulation structures designed with amorphous materials and nanoholes. They developed a nano-thermal cloak utilizing nanophononic metamaterials. Through molecular dynamics simulations, they demonstrated its ability to produce a cloaking phenomenon. The height, number, spacing, and thickness of the base film of the nanopillars significantly influence their cloaking performance. Using a silicon unit cell (UC) as the basic unit, when the base film thickness is 6 UCs, the nanopillar height is 50 UCs, the nanopillar width is 4 UCs, and the nanopillar spacing is 1 UC, the RTC can reach 1.4. The underlying mechanism involves phonon localization in the nanopillar region, which is attributed to the local resonance hybridization effect between the nanopillars and the base film. Furthermore, the nanopillar structure can regulate heat flux through the proximity effect, even in areas without nanopillars.

Design and performance of heat flux concentrators based on different structures. As a practical extension of research on thermal metamaterials, scientists have proposed methods to collect thermal energy by concentrating heat flux. For instance, Dede et al. [252] proposed using thermoelectric generators to collect low-grade waste heat to enhance thermal energy collection. However, the use of a large number of thermoelectric generators in distributed heat sources makes the cost too high. Therefore, it is very advantageous to direct heat to a specific region, so the main goal is to design heat flux concentrators. Zhang et al. [215] designed a heat flux concentrator using an amorphous structure, and the heat flux in the center region can reach 6 times that of the edge region. Further, a heat flux concentrator using a nanohole structure was designed, and the heat flux in the center region can reach 9 times that of the edge region. By calculating the spatial distribution of the localized mode, it is proved that the main reason for the heat flux concentration is the phonon localization in the amorphous and nanohole regions. Zhang et al. [216] designed a heat flux concentrator using nanophononic metamaterials, and proved the heat flux concentration phenomenon by calculating the ratio of heat flux (RHF) between the center and the edge region. The heat flux concentration performance is as high as 1.62. In addition, as the distance from the nanopillar surface increases, the low-frequency peak of PDOS decreases, the high-frequency peak increases, and the MPR changes from localized to delocalized. The effects of the height, spacing, and atomic mass of the nanopillars on the phonon localization effect were also studied. The localization effect produced by the nanopillars is limited to a certain thickness [253]. In heat flux regulation structures, the use of nanopillars grown with heavy atoms can significantly reduce the heat flux. In practical applications, if the configuration of the nanopillar array is fixed, increasing the atomic mass of the atoms in the nanopillars will further reduce the thermal conductivity of the base film. On the other hand, if the volume fraction of the nanopillars to the base film can be increased to a very high value, the effect of the nanopillar configuration will exceed the effect of increasing the atomic mass of the nanopillars. In addition, Anufriev et al. [254] proposed the concept of ray phonons using ballistic heat transport in porous films and experimentally demonstrated the properties of thermal guiding, emitting, filtering, and shielding.

4.3 Future outlook

In terms of fundamental theory, the nano-thermal cloaks and heat flux concentrators developed at this stage primarily focus on thermal fields. However, there is an urgent need for the development of local heat flux regulation structures that are suitable for multi-physical fields. Unlike nanophononic metamaterials, amorphous and nanohole structures can introduce interfacial thermal resistance when designing local heat flux regulation systems, potentially leading to hotspot effects that adversely impact thermal management performance. The heat flux rotation structure can circumvent heat-sensitive devices, allowing for effective heat dissipation. The investigation of nano-thermal rotation structures holds significant engineering importance. Regarding engineering applications, the multi-angle performance optimization of heat flux regulation devices based on nanophononic meatstructures, as well as the exploration of specific engineering challenges, remains insufficient. Currently, research on nano-thermal cloaks and heat flux concentrators is largely confined to theoretical calculations and numerical simulations. Experimental validation of local heat flux regulation structures is essential to advance their engineering applications.

4.4 Summary

This chapter reviews the relevant mechanisms for regulating thermal conductivity through amorphization, nanoholes, and nanophononic metamaterials. It provides a detailed introduction to the development of heat flux regulation structures based on these three approaches. The thermal cloak has significant applications in the thermal isolation of sensitive components. The heat flux concentrator and the thermal cloak serve as complementary structures. The direction of heat flux regulation by the thermal cloak is opposite to that of the heat flux concentrator, allowing heat flux to concentrate in a specific region and maintain an extremely high level of heat flux in that region. This ability can greatly enhance the application of thermoelectric generators. This review presents the design methodology for heat flux regulation structures within nanophononic metastructures, elucidates the heat flux regulation mechanisms, and aims to advance the engineering research of nanophononic metastructures in advanced thermal management.

5 Review chapter on thermoelectricity

Lilia M. Woods^*

　 Department of Physics, University of South Florida, Tampa, FL 33620, USA

5.1 Background

The push for renewable energy in expanding the global energy sector has been a driving force for new research that can result in environmentally friendly applications for competitive economic growth. As a reliable solid-state phenomenon, thermoelectricity is an example of an ecologically friendly and clean method through which heat is directly converted into electricity. It can also operate in the “opposite” mode by using electrical power for cooling and temperature stability [255]. The current thermoelectric technology can produce power generators, Peltier coolers, and embedded solid-state temperature control elements, among others. The global thermoelectric market is expected to reach 1.7 billion dollars by 2027 [256]. The departure from using nonrenewable energy resources motivates vigorous research in the field of thermoelectricity to find better and more efficient devices and energy methods of conversion. This field has been driven forward by synergy between experimental and theoretical studies for the synthesis of materials and device fabrication (Fig.11).

5.2 Theory

Thermoelectric processes are fundamentally related to the laws of thermodynamics. The theoretical description of the coupled electric and thermal flows is achieved within the Onsager−de Groot−Callen theory consistent with the laws of conservation of energy and matter [257, 258]. The theoretical description assumes that the system is driven by a minimum production of entropy, where small perturbations of thermodynamic potentials allow assuming linear coupling between thermodynamic forces and fluxes [259]. This in fact is the definition of irreversible steady-state process near equilibrium conditions for which the quasi-static equation \mathrm{d}S=\frac{\mathrm{d}Q}{T} relates the variation of entropy S and heat Q at a given temperature T. This relation can further be cast into {\mathit{J}}_S=\frac{{\mathit{J}}_Q}{T}, where {\mathit{J}}_S and {\mathit{J}}_Q are the entropy and heat fluxes, respectively. Let us note that entropy plays a central role in the Onsager−de Groot−Callen theory. Thermoelectric processes are irreversible due to the excess production of entropy and they can be conveniently quantified using energetic properties [259].

Starting with the first law of thermodynamics, the complete energy flux is determined by the heat flux and particle flux, such that

(1){\mathit{J}}_E={\mathit{J}}_Q+{\mu }_e{\mathit{J}}_N,

where {\mu }_e is the electrochemical potential. The particle and energy fluxes are associated with forces {\mathit{F}}_N =\mathrm{\nabla }(-\frac{{\mu }_e}{T}) and {\mathit{F}}_E =\mathrm{\nabla }(\frac{1}{T} ) related to the thermodynamic potentials (-\frac{{\mu }_e}{T}) and ( \frac{1}{T}). The linear coupling between the forces and fluxes according to the Onsager−de Groot−Callen theory [260–262] results in constitutive equations, given in Tab.1. The electric, \mathit{J}, and heat, {\mathit{J}}_Q, flux current densities are proportional to the forces via the electric conductivity \overleftrightarrow{\sigma }, thermal conductivity \overleftrightarrow{\kappa }, and Seebeck coefficient \overleftrightarrow{S} (all tensor properties of the material).

The Onsager reciprocity implies that {\overleftrightarrow{\sigma }}^T= \overleftrightarrow{\sigma } and {\overleftrightarrow{\kappa }}^T= \overleftrightarrow{\kappa }. The first term in \mathit{J} represents Ohm’s law for electric transport and the first term in {\mathit{J}}_Q corresponds to Fourier’s law for thermal transport. The thermoelectric coupling is captured by the terms containing the Seebeck coefficient showing that temperature gradient can lead to a charge carrier flux and that the charge carriers can also transport heat flux. These constitutive equations show that indeed entropy is an integral part of the thermoelectric flow. In addition to the thermal part of the entropy flux, there is another contribution of electrochemical origin. In fact, this second part is associated with the Seebeck coefficient measuring the entropy transported per particle charge.

The constitutive equations are consistent with the governing conservation laws. These are also given in Tab.1 and they mathematically express the conserving charge under steady-state conditions (top equation) and the local steady-state energy conservation in the presence of a Joule heat source term (bottom equation). Tab.1 summarizes the mathematical foundation within basic thermodynamic description of thermoelectric transport relating it to specific materials properties.

Major research efforts are directed towards finding materials with specific transport properties that enhance thermoelectricity. Using semiclassical approximation within the Boltzmann transport equation, these characteristics can be related to the electronic structure of the materials [263, 264]. Tab.2 summarizes the most general expressions for the charge transport expressed in terms of the transport distribution \overleftrightarrow{\mathrm{\Theta }}(E). This energy dependent tensorial function dependens on the group velocity \mathit{v}={\hslash }^{-1}\mathrm{\partial }E(\mathit{k})/\mathrm{\partial }k of the charge carriers with corresponding relaxation time tensor \overleftrightarrow{\tau }(\mathit{k}), where \mathit{k} is the wave vector. The energy E(\mathit{k}) captures the energy band structure of the material. The integral expressions for the properties depend on the Fermi distribution function f^0(E)=\frac{1}{{\mathrm{e}}^{E(\mathit{k})-{\mu }_0}+1} ({\mu }_0: chemical potential). These equations constitute the basis of obtaining transport properties by taking into account the electronic structure from first principles simulations.

Thermoelectric devices rely heavily on information about the materials properties. Most commercial applications operate as a thermoelectric heater (TEH), thermoelectric generator (TEG), or thermoelectric cooler (TEC). An applied voltage difference in a TEC or TEC ensures control of the output temperature difference. An applied temperature difference in TEG is used to control the external voltage difference. The efficient operation of these devices relies on general thermodynamic principles [265], and it is typically described by the maximum efficiency of power generation {\eta }^{\mathrm{T}\mathrm{E}\mathrm{G}}, maximum cooling power {\eta }^{\mathrm{T}\mathrm{E}\mathrm{C}}, and maximum coefficient of performance {\varphi }^{\mathrm{T}\mathrm{E}\mathrm{H}} [255, 266], also given in Tab.2.

The performance of the devices is conveniently reduced to maximizing the dimensionless figure of merit Z\overline{T}= \frac{S^2\sigma \overline{T}}{\kappa } composed entirely of the properties of the material. Since these are inter-related, it is rather difficult to find a material with large electrical conductivity and Seebeck coefficient, and low thermal conductivity. It is not possible to change one property without affecting the others, typically in a disadvantageous manner [267].

One of the main goals in the field of thermoelectricity is to design suitable materials to overcome this interdependence. Although there are no fundamental limits on how large the figure of merit can be, it has proven challenging to find materials for thermoelectric devices with competitive efficiency compared to other energy convergence applications. Desired thermoelectric materials are typically semiconductors and the goal is to find and/or design systems with a large power factor PF=S^2 \sigma and small thermal conductivity in order to obtain ZT at least 1.0 or greater, and in recent years, there has been an upward trend of increasing the materials figure of merit, as shown in Fig.12. Several classes of materials have been identified as suitable thermoelectrics, as summarized in what follows.

Materials. Skutterudites and the fillers of their voids can drastically reduce their lattice thermal conductivity, which is perhaps their most attractive thermoelectric feature [301–303]. Their best performance is in the 600−1000 K temperature range. Half-Heusler alloys are cage-like materials, typically for applications above 600 K. Their small band gaps, larger electric conductivity and Seebeck coefficient are key ingredients for their good performance. Multiple atom substitutions and fillers in the atomic structure and relatively small amounts of doping ensure property optimizations that can yield figure of merit ZT>1 [304, 305]. Clathrates are also cage-like materials for which low thermal conductivity can be achieved by the rattling of atoms inserted in their voids. Type I and II clathrates have been studied extensively, although type VII, VIII, and IX families have also been explored [306–308]. The best performing clathrate so far is Ba₈Ga₁₆Ge₃₀ with ZT>1 above 900 K [293] and ZT\sim 0.5 at 773 K [309]. The ternary Zintles are also promising thermoelectric materials due to their strong lattice anharmonicity which limits the lattice thermal conductivity. The tunability of their valence bands is also advantageous to enhance their electric properties [310, 311]. Some of the best performing Zintl compositions with ZT\ge 1 are Yb_0.96Ba_0.004Cd_1.5Zn_0.5Sb₂ and EuCd_1.4Zn_0.2Sb₂ [312, 313]. Si_1−xGe_x alloys have a diamond-like structure and they have also shown very good promise especially in the higher temperature regimes [314, 315]. The thermoelectric performance of Si−Ge alloys can be significantly enhanced by nano-inclusions which reduces the thermal conductivity of the entire system, while doping and defect strategies can further improve the power factor [286, 316–318]. Through microstructure and alloying control n- and p-type materials can easily be achieved.

Organic composites that contain conductive polymers or carbon nanotube solutions with embedded nanostructures made of “traditional” thermoelectric materials are also of interest to thermoelectricity [319, 320]. The extremely low thermal conductivity from the organic/carbon matrix combined with the advantageous electric properties of the inorganic additions of Bi₂Te₃ and Sb₂Te₃ gives very promising results in terms of large figure of merit at lower temperatures (500 K) [312, 321]. Binary chalcogenides, such as IV-VI compounds (PbTe, SnSe, GeTe, etc.) and V−VI compounds (Bi₂Te₃, Sb₂Te₃, Bi₂Se₃, etc.) in the rock-salt structure are especially promising for room temperature thermoelectric conversions [322]. In fact, the majority of applications are made of bismuth-antimony or selenium-telluride systems. Doping, defects, alloying, and other strategies result in an advantageous balance of a low thermal conductivity and enhanced power factor due to successful band engineering [323–326]. Ternary and Quaternary Chalcogenides derived from binary II−VI, IV−VI, and V−VI parent compositions have also recently attracted interest in the TE community [327–329] due to their inherently low thermal conductivity. Strategies for enhancing and optimizing their electric transport are also being explored.

Recent progress on novel and efficient TE materials is driven by advanced simulations methods. Density functional theory (DFT), molecular dynamics, and nonequilibrium Green’s function theory have been successfully used for materials properties and transport mechanisms predictions [330, 331]. High-throughput calculations for accelerated screening of materials have helped create databases of compositions with promising thermoelectric properties [332, 333], which are now actively being used in machine learning methods for thermoelectric informatics [334]. Predictions using elemental characteristics [335], and active learning machine learning frameworks combined with DFT and experiments are at the forefront for computational design of more efficient materials [336–338].

Device applications. Typical devices exploiting thermoelectric conversion phenomena are modules based on state of the art materials with structural and thermal stability. Application of thermoelectric power generators and refrigerators are in areas where the main requirement is the reliability of the device and the efficiency is of secondary priority [339]. Automobile companies, such as Honda, Toyota, and BMW, are implementing thermoelectric generators to recapture some of the wasted heat by the engine exhaust back into the combustion cycle boosting the overall efficiency of the car [340, 341]. Radioisotope thermoelectric generators are widely used in space exploration expeditions because of their superior reliability as compared to other methods. The heat from the radioactive materials undergoing natural decay is used to generate electricity [342]. Recently, thermoelectric modules have found applications in a variety of wireless sensor networks and medical devices. For example, thermoelectric powered systems have entered the market for low-cost environmental sensing [343, 344] and medical devices [345, 346]. Thermoelectric junctions are now employed for novel photodetectors based on selective resonant nanophotonics [347].

Beyond single materials. Finding materials with superior thermoelectric properties via experimental and theoretical means can be viewed as a microscopic research approach. Nevertheless, macroscopic methods are also being used to design systems with targeted functionalities. Using scattering techniques or transformation optics different types of metamaterials have been proposed not only to increase the efficiency of the thermoelectric conversion processes, but also to manage thermoelectric flow.

Phononic nanocrystals, as previously discussed, are of great interest to the field of thermoelectricity as potentially useful platforms to reduce the thermal conductivity while not affecting the electronic transport significantly. Such composite systems are also useful for thermal flow management applications. Phononic nanocrystals contain an arrangement of structures within a substrate to engineer phonon dispersions. The periodicity of this arrangement has a unit cell comparable to the phonon wavelength [348].

An array of intriguing routes have been proposed to engineer phonons with such microstructures. The theoretical descriptions rely primarily on simplified models for obtaining characteristic frequencies of coupled oscillators or continuum models of acoustic wave propagation including nonlinearity [347]. Continuous models supplemented by periodicity of the lattice have been tackled via numerical techniques, such as finite difference, multiple scattering, and plane wave expansion methods for more accurate predictions. For thermoelectricty in particular, phononic nanocrystals composed of embedded nanoparticles lead to a resonance effect altering the phonon spectrum due to hybridization between the nanoparticles local resonances and the underlying atomic lattice dispersion [349–351]. The main benefits of resonant phononic crystals comes from blocking especially the the lower frequency range of phonon transport, which is typically unaffected by other nanostructures. Experimental studies have shown that indeed it is possible to achieve in practice a phononic resonant crystal which significantly reduces \kappa without affecting the power factor [352].

Composite materials for controlling the electronic flow to enhance the power factor in the figure of merit have also been proposed. Core−shell nanoparticles embedded in a semiconducting host can become invincible to the flow of electrons with a specific energy range giving a boost to the overall carrier mobility [353, 354]. The double barrier from the coated nanoparticles gives the flexibility to achieve such a cloaking effect, which is difficult to obtain otherwise. Scattering theory via partial wave decomposition was instrumental in revealing that electronic scattering cross section from a double barrier of the core-shell nanoparticle is significantly reduced. This anti-resonant effect assures high carrier concentration in the host material with practically invincible nanoparticles for the electrons. On the other hand, phonons experience an additional scattering mechanism reducing the thermal conductivity further benefiting the thermoelectric conversion. Experimental evidence of the enhanced figure of merit due to nanoinclusions in various material hosts has been reported already [355, 356]. Recently, it has been shown that the collective electronic cloaking from a cluster of core−shell nanoparticles is also possible [357, 358]. The nearly complete suppression of forward and backward scattering demonstrated theoretically via partial wave scattering composites brings new directions for core-shell cloaks involving higher concentration of nanoparticles in the host material. Such clusters are envisioned to have phonon-blocking and electron transmission enhanced capabilities.

Transformation optics. Recent ideas based on thermodynamic principles to manipulate the thermoelectric transport at the macroscopic scale have been fueled by the field of optics, in which exotic phenomena such as electromagnetic fields cloaking or metamaterials with negative index of refraction among others have been made possible [359, 360]. The idea of transformation optics used to construct optical metamaterials can also be applied to diffusive phenomena that have similar functionalities as in optics, but the control is over mass and energy flows [361, 362]. Experimental studies guided by theory give encouraging evidence that heat flow cloaking, focusing, and reversal are indeed achievable in the laboratory [363–365].

The mathematical background of transformation optics can be traced to topology in which vector fields in a given region can be mapped to another subregion by maintaining the identity mapping on its boundary [366]. This mapping is captured mathematically by a coordinate transformation Jacobian matrix A_{ij} =\frac{\mathrm{\partial }{\mathit{r}}_i^{\prime }}{\mathrm{\partial }{\mathit{r}}_j}, where {\mathit{r}}^{\prime },\mathit{r} are position vectors in the two regions. The key feature is that the transformation preserves the form invariance of the governing laws and constitutive equations for a given set of physical phenomena.

The application of the transformation optics to the governing and constitutive thermoelectric equations obtained directly from thermodynamics [367, 368] results in thermoelectric materials properties, that are highly anisotropic as implied by the Jacobian matrix (see Tab.1). This is the case even if the original materials was completely isotropic. Nevertheless, this is a general method that is entirely determined by the type of the coordinate transformation one is specifying. This method does not place any requirements on initial and boundary conditions, geometry, or any other characteristics of the region being transformed. It is this generality that allows transformation optics to be applied to multi-physics problems with many types of flow control. In the case of thermoelectricity, cloaking, rotating, concentrating, diffusing of the coupled via the Seebeck coefficient electric and heat flows have been proposed [367–371]. The practical implementation of such unusual manipulations of thermoelectricity requires materials with extreme properties. For example, cloaking of a circular region is possible if the thermal and electrical conductivities are infinite in the azimuthal direction, but zero in the radial direction. Since such materials are not available in nature, it has been suggested that constructing metamaterials following in-parallel and in-series connected elements from basic circuitry can realize such effects in practice.

5.3 Future outlook

The foundation of thermoelectricity lays in the principles of thermodynamics. Basic research in this field is especially motivated by the practical implications of an environmentally friendly method of energy conversion. Most of the progress has been in the search of materials with desired transport properties to be used for the efficiency improvement of current devices. However, management of thermoelectric flow is also gaining much interest [372]. A wearable camouflage blanket makes a person invisible to heat flow [373], while a Janus cloak ensures car battery protection against severe temperature swings [374]. Thermoelectric pocket airconditioners by SONY are also available in the market.

Robust methods to design metamaterials that can control thermoelectric flow are needed for basic understanding of transport in composite materials and applications [362]. Transformation optics provides a reliable route for managing diffusive flows with multifunctional purposes. Much of the work currently is primarily theoretical, thus experimental realization for a broader range of predicted systems is needed.

Research focusing on materials is expected to grow, especially with the wider application of machine learning methods and the creating of reliable materials repositories. A novel idea proposing network theory for inhomogeneous materials [375] can improve comparisons with experiments, but it also has the potential for reliable modeling of disordered bulk materials. This idea can further be explored by machine learning or other AI methods in order to examine the possibility for the construction of inhomegenous systems for improved thermoelectric properties.

Another direction of interest is the transverse thermoelectric effect in materials since a large transverse effect facilitates thermoelectric modules with higher efficiency [376, 377]. Dirac semimetals, Weyl semimetals, and nodal line semimetals and their nontrivial band topology are emerging as systems with substantial transverse thermoelectric effect. Synergistic theory-experiment collaborations are much needed to advance fundamental understanding of this type of thermoelectricity.

6 Transformation thermotics

Gaole Dai^*

　 School of Physical Science and Technology, Nantong University, Nantong 226019, China

6.1 Background

Thermal metamaterials made of artificial structures can regulate macroscopic heat transfer efficiently, enabling functionalities beyond most natural materials [378–380]. The challenge lies in how to determine the geometric structure and material parameters (such as thermal conductivity, density, and specific heat) based on the desired regulation function. Transformation theory (referred to as transformation thermotics or transformation thermodynamics in the context of thermal metamaterials) provides a fundamental analytical approach for addressing such inverse problems. The origins of transformation theory can be traced back to Refs. [381–383], where its establishment in the field of optics [384, 385] over the past two decades has significantly propelled the development of metamaterials. It has since been applied to fields such as acoustics, thermodynamics, and fluid mechanics [386, 387].

The starting point of transformation theory is that the governing equations of some physical systems can maintain their forms under arbitrary coordinate transformations [384]. Governing equations are usually expressed by tensors and their low-rank counterparts, i.e., vectors and scalars. The objectivity of tensors ensures that the governing equations must apply in all coordinate systems. However, the specific expressions of governing equations written by tensor components may vary between coordinate systems. The so-called form invariance actually requires that the expressions differ only in their coefficients, without introducing additional terms, such as affine connections, unless they can be explained through extra physical mechanisms. The non-trivial aspect of transformation theory lies in establishing a correspondence between the physical fields in two different spaces. For example, the governing equation in a curvilinear coordinate system for a homogeneous and isotropic space can take the same form as that for an inhomogeneous and anisotropic space in a Cartesian coordinate system. In this way, if the physical fields and material properties of one space (called the virtual space) are known, and the desired physical field distribution in another space (called the physical space) can be mapped from the virtual space through a coordinate transformation, then the material properties in the physical space can be calculated based on the information of coordinate transformation. This idea is analogous to the principles of general relativity [388], where the physical fields (analogous to the path of light) are manipulated through material properties such as permittivity and permeability, which are comparable to the curvature of spacetime.

6.2 Past to current development

6.2.1 Establishment and early developments

Heat transfer occurs in three fundamental modes: conduction, convection, and radiation. Transformation thermotics was initially developed within the framework of heat conduction [389]. Unlike electromagnetic waves, heat conduction is a diffusion process, and the macroscopic heat conduction governed by the Fourier’s law does not integrate as seamlessly with general relativity as Maxwell’s equations do. In fact, it even conflicts with special relativity, as the Fourier’s law hints that the speed of heat propagation is infinite. However, heat conduction still satisfies the form invariance required by the transformation theory. Studies in 2008 [389, 390] and 2011 [391] respectively studied transformation thermotics in both steady-state and transient scenarios, and these theoretical predictions were subsequently validated by experimental results [392, 393].

Early studies in transformation thermotics demonstrated functionalities such as thermal cloaking, heat flux concentration, and heat flux rotation, based on two-dimensional transformations [391, 392]. Similar to metamaterials in other fields, the thermal conductivity derived directly from transformation theory are usually anisotropic and spatially inhomogeneous. In transient scenarios, inhomogeneous distributions of density and specific heat are also required. Linear transformations can bypass the inhomogeneity of thermal conductivity, but eliminating anisotropy is more challenging, which will be discussed in detail below. Experimentally, these functionalities were approximated using composite materials since anisotropic conductivity is often beyond the property of natural materials. For instance, concentric ring structures were used to achieve thermal cloaking, fan-shaped structures for heat concentration, and spiral structures for heat flux rotation [392, 393].

6.2.2 New coordinate transformations

During the last decade, transformation thermotics has made significant advancements in both theoretical and practical aspects. The control of thermal fields directly depends on how the coordinate transformations map the physical fields from the virtual space to the physical space. The first area of progress, therefore, involves a broader variety of coordinate transformations. For example, thermal focusing for convergent conduction were achieved by a compound of translation, rotation, and stretch-compression [394]. Other important advances in new coordinate transformations include the introduction of transforming heat source. Thermal illusions were realized by transforming the heat source into multiple splits or by displacing the heat source [395, 396]. What is more, through nonlinear folding compression of boundaries, researchers proposed active (i.e., with elaborately-distributed heat sources) thermal metasurfaces behaving like a thermal lens and even showing an effective negative thermal conductivity [397] [Fig.13(a)]. Three-dimensional coordinate transformations are also being used more frequently to design three-dimensional metamaterials, such as a thermal spreader preferentially guiding the heat flux [398], a metadevice that exhibits different functions depending on the direction of heat flux [399] [Fig.13(b)], and a meta‐helmet for wide-angle thermal camouflages [400].

Additionally, two types of special coordinate transformations that induce totally different degrees of anisotropy have recently garnered the attention of researchers. Most coordinate transformations lead to anisotropic material properties in the physical space (referred to as the transformation media), which makes fabrication challenging. Reducing or even eliminating the anisotropy of transformation media could greatly simplify experiments without losing the function. It is well known that conformal mappings do not introduce additional anisotropy, nor do they change the thermal conductivity after the transformation. However, the Cauchy−Riemann conditions required by conformality are too restrictive, making it difficult to achieve flexible control functions. Researchers have recently introduced the concept of pseudo-conformal mapping [401], an extension of conformal and quasi-conformal mappings [402] in the two-dimensional space. In isotropic media, heat flux streamlines and isotherms are always orthogonal based on the Fourier’s law. Pseudo-conformality can preserve this orthogonality. Although the thermal conductivity derived from traditional transformation theory is anisotropic, the isotherms and streamlines are aligned with the principal axes of the thermal conductivity tensor. This guarantees that we only need the principal values of the thermal conductivity along the flux lines to achieve the desired function, thereby avoiding the anisotropy. Using this approach, researchers have successfully realized functions such as heat flux guiding and expanding [401]. Additionally, they discovered that the bilayer cloak [403, 404] and the zero-refraction cloak [405], previously outside the framework of transformation theory, could be derived using pseudo-conformal mappings [401] [Fig.13(c)]. Further research proposed the method of “forward conformality-assisted tracing” [406] in the three-dimensional spaces, offering an effective and analytical tool in thermal control and material design.

On the other hand, coordinate transformations can also lead to extremely anisotropic transformation media. The most extreme anisotropy, for example, in the two-dimensional case, means that the off-diagonal components of thermal conductivity can be neglected, while the two diagonal components are infinity and zero, respectively. This type of transformation media is homogeneous and has been utilized in recent studies, referred to as “transformation-invariant metamaterials” [407, 408] or “thermal null media” [409, 410] [Fig.13(d); “null” means the width \mathrm{\Delta } in the virtual space is close to zero]. Usually, the parameters of transformation media depend on both the coordinate transformation and the environment, whereas transformation-invariant metamaterials can maintain their parameters when the environment changes, ensuring that the original design will not fail. This property can be utilized to realize intelligent metamaterials that behave like chameleons [408], which can adapt to their environment. Furthermore, the feasibility of transformation-invariant metamaterials are also independent of the required device geometry [409, 410]. This reconfigurability can facilitate the rapid design of arbitrarily shaped metamaterials without complicated parameter calculations.

6.2.3 New scenarios

The second area we are reviewing here extends the scope of transformation thermotics beyond simple heat conduction to more complex scenarios, for example, establishing the transformation theory to spatiotemporal media [411], considering the effects of convection in the environment in experiments of conductive metamaterials [412], and utilizing surface infrared radiation for thermal camouflage [413] [Fig.14(a)]. The simultaneous control of thermal fields with other physical fields (such as electrostatic fields [414–417] and electromagnetic waves [418, 419]) has also attracted widespread interest, as it can achieve the same [414–419] or different [417] control functions for the two physical fields. For example, Chen et al. [419] realized a thermal-electromagnetic concentrator able to amplify the heat flux and electromagnetic signals simultaneously [Fig.14(b)]. In the aforementioned studies, there is no direct interaction between the thermal field and other physical fields; they are decoupled. Stedman and Woods [420] investigated the coupled thermal and electric fields, specifically the transformation theory under the thermoelectric effect, and found that the governing equations still possess form invariance under coordinate transformations. Based on their finding, it might be possible to enhance the thermoelectric power generation by adjusting the temperature distribution via a concentrator.

The mechanical motion of media can lead to another mode of heat transfer beyond conduction, namely advection or convection. This conduction−advection model also meets the requirements of transformation theory. Dai et al. [421] introduced the Darcy model for porous media to generate the flow velocity of the medium and established a transformation thermal convection model. By adjusting the contributions of conduction and advection to heat transfer, Jin et al. [422] discovered that changing the flow velocity allows metamaterials to switch between thermal invisibility and thermal concentration, achieving a kind of topological transition [Fig.14(c)]. The mechanical motion of the medium can also be triggered by variations in the thermal field, such as the phase transition of shape memory alloys (SMAs), and has been used in nonlinear transformation thermotics (where the material properties are temperature-responsive) to realize switchable thermal invisibility [423, 424]. Recently, Chen et al. [425] proposed a force−structure coupling mechanical system made of rotatable unit cells driven by nested gears, to achieve continuously tunable functions by actively rotating the unit cells when external conditions change. These advances in intelligent metamaterials can benefit the increasing demand of thermal management, e.g., in integrated circuit packaging.

The last fundamental mode of heat transfer, namely thermal radiation, can be equivalently represented as a nonlinear thermal conductivity (cubic in temperature) under the Rosseland approximation for optically thick media. This can also be incorporated into the framework of transformation thermotics to achieve simultaneous control of conduction and radiation, which can be utilized to enhance the thermal protection performance of high-speed aircraft [426].

6.2.4 High-order applications

In the previous sections, we reviewed some theoretical advancements in transformation thermotics, mainly focusing on the application of new coordinate transformations and the implementation of specific control functions in new heat transfer modes. To more effectively utilize the abundant low-grade thermal energy in the environment and achieve higher-order functions, such as information processing and transmission, it is necessary to integrate and design transformation media with different functionalities as a cohesive system. For instance, Hu et al. [427] proposed the concept of thermal coding under macroscopic heat conduction. They defined the effects of invisibility and concentration as binary codes 0 and 1, respectively, allowing these two types of transformation media to be arranged as distinct basic units; see Fig.15(a). This arrangement enables the output of thermal signals under an external heating field, showing a potential for logic calculation. Later, Lei et al. [428] used temperature-responsive media to realize a temperature-controlled all-thermal encoding strategy based on the temperature-dependent transformation thermotics. In addition, the time dimension was introduced into the scheme of thermal encoding via a spatiotemporally modulated metashell [429]. Another integrated high-order function is thermal encrypted printing [430], which, through the stitching of several basic coordinate transformations in illusion thermotics, can produce images of the English alphabet in infrared detection, thereby serving as a method of information transmission; see Fig.15(b).

6.3 Future outlook

Despite the encouraging advancements in transformation thermotics, we can still anticipate future breakthroughs in theory and practical implementations. In terms of theoretical research, the coordinate transformations used in transformation thermotics are currently limited to spatial coordinates, with little exploration of time-coordinate transformations, which has been done in the theory of optical transformations, called “spacetime cloaks” [431]. This limitation restricts the control of transient heat transfer. Moreover, when both space and time coordinates are transformed, or when frame transformations are considered, more intriguing physical phenomena could emerge. In the optical counterpart of transformation theory, researchers have successfully investigated phenomena such as non-Hermitian effects [432, 433], chirality [434], non-reciprocity [435], topological transitions [435], and parity−time symmetry [437] through coordinate transformations. Although these phenomena have also garnered significant interest in thermotics recently, their connection to coordinate transformations has yet to be uncovered, making it a promising area for further exploration. Additionally, transformation thermotics directly informs us of the spatial distribution of material properties needed to achieve specific control functionalities. In practical applications, natural materials rarely meet the exact parameter requirement of transformation media, making it necessary to design composite materials to realize these functionalities. Numerical optimization methods, on the other hand, allow for the inverse design of the composite structure directly from the desired control function. With the rise of AI for science, integrating the insights of transformation thermotics into algorithms could provide valuable prior knowledge for the AI-driven design of metamaterials, making this a promising area for future research [438].

6.4 Summary

In summary, we have reviewed the establishment and development of transformation thermotics since 2008 and provided an outlook on future research directions. Transformation thermotics has employed an increasingly diverse range of coordinate transformations, including extensions from two dimensions to three dimensions, incorporating the effect of heat sources, the use of generalized conformality to avoid anisotropy, and the implementation of homogeneous extreme anisotropy to achieve adaptability to environmental changes and geometric reconfigurability. In addition, its scope has expanded from heat conduction to more complex and realistic heat transfer modes. Benefited from these advances, the control functions have evolved from simple operations to more advanced and high-order capabilities, including switchable, adaptive, and intelligent thermal regulation. As an analytical approach to designing thermal metamaterials from a geometric perspective, transformation thermotics is expected to theoretically bridge other areas of thermal metamaterials and find applications in more practical scenarios.

7 Thermal metamaterials based on the temperature dependent transformation thermotics

Shuzhe Zhang², Xiangying Shen^1,*

　 ¹School of Science, Shenzhen Campus of Sun Yat-sen University, Shenzhen 518107, China

　 ²Department of Materials Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China

7.1 Background

Over the past decade, transformation mapping theory has gained recognition as a framework for designing metamaterials [439–442]. Research has demonstrated the feasibility of creating thermal metamaterials using this approach [443]. However, the inherent complexity and diffusivity of heat flow complicate its control. Traditional materials have proven inadequate for precise heat management, driving interest in metamaterials as a solution. The intricacies of thermodynamics further challenge the study of these metamaterials. Building on transformation optics, researchers developed transformation thermotics to guide thermal metamaterial design. This theory enables functions such as heat concentration, convergence, and rotation, leading to classifications like thermal concentrators, cloaks, rotators, and lenses. The theoretical foundation of transformation thermotics lies in coordinate transformation, central to the design methods discussed here.

As discussed earlier, the concept of optical cloaking has inspired the development of various metamaterials. In 2006, Pendry et al. [444] and Leonhardt et al. [445] independently designed optical cloaks, introducing a new paradigm for metamaterial design [446–448]. The foundation of this theory, known as transformation optics, is based on coordinate transformation, which alters the propagation path of light, allowing it to bypass cloaked objects and achieve optical invisibility.

In 2008, building on earlier research, studies [449, 450] established the theory of steady-state transformation thermotics. This work introduced the concept of a thermal cloak, predicted the thermal cloaking effect, and discussed apparent negative thermal conductivity. In that same year, Chen et al. [450] applied coordinate transformation methods to create a thermal cloak, exploring its potential in anisotropic environments and further validating the use of coordinate transformations in designing thermal metamaterials.

In 2015, Li et al. [451, 452] examined a generalized heat conduction process characterized by temperature-dependent thermal conductivity. This research led to the development of a new theoretical frameworktemperature-dependent transformation thermodynamics. Unlike previous models, this nonlinear theory accommodates variations in thermal conductivity with temperature, enabling the design of devices that utilize materials with adaptive thermal conductivities for advanced heat flow manipulation. They proposed a macroscopic thermal diode, presenting a novel approach to thermal energy management. Additionally, they theoretically designed and experimentally validated a thermalstat that maintained a stable temperature within its core, even under significant change of the external temperature gradients.

7.2 Theory

In 2015, Li et al. [451] proposed the temperature dependent transformation thermotics, which considers a generalized thermal conduction process where thermal conductivity varies with temperature, culminating in the development of a new theoretical framwork of transformation thermotics.

The theoretical foundation is represented by the generalized Fourier law governing steady-state heat conduction in the absence of thermal sources:

(2)\mathrm{\nabla }\cdot \left[\kappa \left(T\right)\cdot \mathrm{\nabla }T\right]=0.

Here, \kappa \left(T\right) denotes a temperature-dependent function. Integrating nonlinear materials into the transformation thermodynamics framework necessitates demonstrating the invariance of Eq. (2) under coordinate transformations; such invariance is crucial for accurately calculating the thermal conductivity tensor in transformed spaces.

In an n-dimensional curved coordinate space characterized by covariant basis \left(g_1,g_2,\dots ,g_n\right), the component form of Eq. (2) can be expressed as

(3){\mathrm{\partial }}_i{\kappa }^{ij}\left(T\right){\mathrm{\partial }}_{j}T+{\mathrm{\Gamma }}_{ik}^i {\kappa }^{kj}\left(T\right){\mathrm{\partial }}_{j}T=0.

Here, {\mathrm{\Gamma }}_{ik}^i represents the Christoffel symbol, defined as the i-th contravariant component in the differential of coordinate x^i over g_k, while {\kappa }^{ij} is the ij-th contravariant component of the thermal conductivity tensor in the specified space. The Einstein summation convention is applied, allowing for the implicit summation of indices.

Eq. (3) can be simplified by introducing the metric tensor G, with component definitions provided as g_{ij}=g_i\cdot g_j. The Christoffel symbol can then be expressed as

(4){\mathrm{\Gamma }}_{ik}^i= {\displaystyle \frac{1}{2}}{g}^{il} {\mathrm{\partial }}_k{g}_{il}={\displaystyle \frac{1}{\sqrt{g}}}{\mathrm{\partial }}_k \sqrt{g}.

In this equation, g denotes the determinant of the matrix g_{ij}. Subsequently, the component form of Eq. (2) is reformulated as

(5){\mathrm{\partial }}_i\left[\sqrt{g}{\kappa }^{kj}\left(T\right){\mathrm{\partial }}_{j}T\right]=0.

In a curved coordinate system, the matrix representation of the contravariant basis \left(g_1,g_2,\dots ,g_n\right) corresponds to the Jacobian matrix J for the transformation between physical space and curved space, allowing us to express:

(6)\sqrt{g}=\sqrt{\mathrm{d}\mathrm{e}\mathrm{t}\left(J^{-\mathrm{T}}{J}^{-1}\right)}=\mathrm{d}\mathrm{e}\mathrm{t}\left(J^{-1}\right)={\mathrm{d}\mathrm{e}\mathrm{t}}^{-1}\left(J\right).

Upon transitioning from Cartesian coordinates x^i to new coordinates x^{\mathrm{\prime }i}, the expression transforms into

(7)\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}\frac{{\kappa }^{ij}\left(T\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\left(J\right)}\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}^j}=& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }k}}\frac{\mathrm{\partial }{x}^{\mathrm{\prime }k}}{\mathrm{\partial }{x}^i}\frac{{\kappa }^{ij}\left(T\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\left(J\right)}\frac{\mathrm{\partial }{x}^{\mathrm{\prime }l}}{\mathrm{\partial }{x}^j}\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}^{\mathrm{\prime }l}}\\ =& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mathrm{\prime }k}}\frac{J_{\mathrm{i}}^{\mathrm{k}}{\kappa }^{ij}\left(T\right)J_{\mathrm{j}}^{\mathrm{l}}}{\mathrm{d}\mathrm{e}\mathrm{t}\left(J\right)}\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}^{\mathrm{\prime }l}}=0.\end{array}

The above equations reveal two critical aspects: first, the form of the equation remains invariant under coordinate transformations; second, the thermal conductivity tensor \tilde{\kappa }\left(T\right) in physical space and the temperature-independent thermal conductivity tensor \kappa \left(T\right) in the new coordinate space share identical forms:

(8)\tilde{\kappa }\left(T\right)={\displaystyle \frac{J\kappa \left(T\right)J^{\mathrm{T}}}{\mathrm{d}\mathrm{e}\mathrm{t}\left(J\right)}}.

In this context, \tilde{\kappa }\left(T\right) and \kappa \left(T\right) represent the matrix forms of thermal conductivity. This foundational work presents the effective transformation theory for temperature-dependent transformation thermodynamics, enabling the design of nonlinear thermal metamaterials.

To extend the applicability of this framework to temperature-related changes in thermal conductivity, Eq. (8) can be reformulated as follows:

(9)\tilde{\kappa }\left(T\right)={\displaystyle \frac{\tilde{J}\left(T\right){\kappa }_0\left(T\right){\tilde{J}}^{\mathrm{T}}\left(T\right)}{\mathrm{d}\mathrm{e}\mathrm{t}\left[\tilde{J}\left(T\right)\right]}},

where {\kappa }_0 is a background thermal conductivity independent of temperature. For detailed proofs, please refer to the supplementary materials of Li et al. [451].

In conclusion, we have introduced the fundamental theory of thermal metamaterials based on the temperature dependent transformation thermotics, emphasizing that the dependence of thermal conductivity on temperature can be integrated through temperature-dependent transformations. This theoretical foundation paves the way for the design of thermal metamaterials that utilize temperature transformations, which will be elaborated upon in the following section.

7.3 Application

Below, we introduce three thermal metamaterials or devices devised based on the temperature dependent transformation thermotics: switchable thermal cloaks and macroscopic thermal diodes, thermostat without energy consumption, thermal cloak-concentrator.

7.3.1 Switchable thermal cloaks and macroscopic thermal diodes

Li et al. [451] proposed switchable thermal cloaks, where in the thermal response of the cloak varies based on its temperature. The concept involved designing two distinct thermal cloaks integrated into a single device. Two cloaks were initially constructed: Type A (active at high temperatures) and Type B (active at low temperatures), each losing their cloaking properties outside the designed temperature range. By integrating these, a macroscopic thermal diode was achieved, providing differential thermal responses based on the direction of heat flow.

To achieve switchable cloaking, a standard cloak maintains a constant central region temperature, shielding it from external heat while not disturbing the external thermal distribution. This effect can be realized using a simple radial stretching transformation in polar coordinates, compressing a region of radius R_2 into an annular region between R_1 and R_2:

(10)r^{\mathrm{\prime }}=r{\displaystyle \frac{R_2-R_1}{R_2}}+R_1.

For switchable cloaking, where heat flow exhibits different directional behaviors, two types of thermal cloaksdenoted \alpha and \beta are combined. Th \alpha-type cloak maintains cloaking at high temperatures, losing this effect at low temperatures, while the \beta-type cloak exhibits the opposite behavior. Thus, cloak functionality is switched based on external temperature conditions, enabling adaptive thermal responses. To facilitate this switchable mechanism, the transformation equation is modified:

(11)r^{\mathrm{\prime }}=r{\displaystyle \frac{R_2-{\tilde{R}}_1\left(T\right)}{R_2}}+{\tilde{R}}_1\left(T\right).

In this equation, {\tilde{R}}_1\left(T\right) is introduced to render the transformation temperature-dependent. Specifically, for the \alpha-type cloak: {\tilde{R}}_1\left(T\right) =R_1\left[1-{\left(1+{\mathrm{e}}^{\beta \left(T-T_C\right)}\right)}^{-1}\right], for the \beta-type cloak: {\tilde{R}}_1\left(T\right) =R_1/\left[1+{\mathrm{e}}^{\beta \left(T-T_C\right)}\right], where T_C is the phase transition temperature that determines the activation state of the cloak, and {\beta }_0 is a scaling factor controlling the rate of the phase transition, set to 2.5 {\mathrm{K}}^{-1} in the study [451].

Using the above relationship, the thermal conductivity tensor in polar coordinates for the switchable cloak can be expressed as diag\left[{\tilde{\kappa }}_r\left(T\right),{\tilde{\kappa }}_{\theta }\left(T\right)\right]:

(12)\begin{array}{rl}& {\tilde{\kappa }}_r\left(T\right)={\kappa }_0\frac{r^{\mathrm{\prime }}-{\tilde{R}}_1\left(T\right)}{r^{\mathrm{\prime }}},\\ & {\tilde{\kappa }}_{\theta }\left(T\right)={\kappa }_0\frac{r^{\mathrm{\prime }}}{r^{\mathrm{\prime }}-{\tilde{R}}_1\left(T\right)},\end{array}

where {\kappa }_0 represents the temperature-independent thermal conductivity of the background material.

The thermal conductivity tensor is temperature-dependent, non-uniform, and anisotropic, complicating experimental realization. They proposed achieving this by alternating two homogeneous, isotropic materials with temperature-dependent properties.

With layer thicknesses d_1=d_2 and thermal conductivities {\kappa }_a,{\kappa }_b, effective medium theory dictates that {\kappa }_a+{\kappa }_b\approx {\kappa }_0^2 to maintain a consistent heat field. By making {\kappa }_a and {\kappa }_b temperature-dependent, conductivities {\kappa }_1\left(T\right) and {\kappa }_2\left(T\right) are obtained:

(13)\begin{array}{rl}& {\kappa }_1\left(T\right)={\kappa }_a+\frac{{\kappa }_0-{\kappa }_a}{1+{\mathrm{e}}^{\beta \left(T-T_C\right)}},\\ & {\kappa }_2\left(T\right)={\kappa }_b+\frac{{\kappa }_b-{\kappa }_a}{1+{\mathrm{e}}^{\beta \left(T-T_C\right)}}.\end{array}

For temperatures T\gg T_C, {\kappa }_1\left(T\right)\to {\kappa }_a and {\kappa }_2\left(T\right)\to {\kappa }_b, while for T\ll T_C, {\kappa }_1\left(T\right)\to {\kappa }_0 and {\kappa }_2\left(T\right)\to {\kappa }_0. Thus, above the phase transition temperature, the cloak functions as intended, while at lower temperatures, it becomes transparent by matching the background conductivity.

A method for constructing macroscopic thermal diodes was also proposed by Li et al. [451], where \mathrm{\aleph } and \beta thermal cloaks are placed at specific positions (I and II) with a conventional conductor in region III [Fig.17(B)]. The differing temperature responses of the cloaks result in distinct thermal distributions based on heat flow direction. Fig.17(B) demonstrates that the diode allows heat flow from left to right while significantly restricting it in the reverse direction, resulting in a rectification ratio of up to 30, depending on the parameter settings.

The proposed macroscopic thermal diode was constructed using shape memory alloys (SMAs) to enable thermal conductivity modulation via structural deformation. Alternating copper and foam plastic layers, as shown in Fig.17(C), were used for the cloak. Near the phase transition temperature, the SMA deforms, altering the local thermal conductivity by connecting or disconnecting copper sheets. In the insulated state, minimal heat flow is observed, while in the conductive state, a substantial thermal gradient appears internally. The temperature distribution aligns with Fourier’s law, where the temperature gradient correlates with heat flux density given identical thermal conductivities.

In summary, they proposed a nonlinear transformation thermodynamics framework and introduced a macroscopic thermal diode design. These thermal diodes [453] demonstrate significant rectification effects and hold potential applications in thermal protection, heat dissipation, and thermal illusions, offering a novel direction for future research. By effectively employing temperature-dependent nonlinear transformations, metamaterials with tunable thermal rectification capabilities can be realized.

7.3.2 Thermostat without energy comsuption

Heat is ubiquitous in nature, but effective utilization remains challenging. Uncollected heat is ultimately wasted. Based on nonlinear transformation thermotics, Shen et al. [454] proposed a temperature-trapping theory using phase-change materials with temperature-responsive thermal conductivity. They theoretically designed and experimentally validated a novel thermostat capable of maintaining a constant temperature without external energy input, even under fluctuating environmental temperature gradients. As an application, the authors designed and simulated an insulating thermal cloak that maintains a constant temperature in its central region despite significant changes in ambient temperature gradients, setting it apart from conventional thermal cloaks.

The concept of an energy-free thermostat was first introduced. It is well known that thermal conductivity depends primarily on temperature [439], and it undergoes abrupt changes during phase transitions [455]. To simplify the analysis, onedimensional steady-state heat conduction along the horizontal axis is considered, with thermal conductivity as a function of temperature:

(14)\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}x}\left[\kappa \left(x,T\right)\frac{\mathrm{d}T}{\mathrm{d}x}\right]=\frac{\mathrm{\partial }\kappa \left(x,T\right)}{\mathrm{\partial }x}\frac{\mathrm{d}T}{\mathrm{d}x}\\ & +\frac{\mathrm{\partial }\kappa \left(x,T\right)}{\mathrm{\partial }T}{\left(\frac{\mathrm{d}T}{\mathrm{d}x}\right)}^2+\kappa \left(x,T\right)\frac{{\mathrm{d}}^{2}T}{\mathrm{d}{x}^2}=0.\end{array}

Next, a specific function must be found that ensures that the temperature of a specific region remains (nearly) constant, even with significant changes in boundary conditions. In other words, when the external temperature field changes, the material should maintain a stable internal temperature. Considering a one-dimensional model with a heat source at the right end (temperature T_H) and a cold source at the left end (temperature T_L), we set an intermediate temperature T_C between T_H and T_L at the midpoint of the model, as shown in Fig.18. This divides the model into two regions: T_H to T_C and T_H to T_L. If T_H changes while \kappa \left(x,T\right) remains the same, the temperature in the second part must always range from T_C to T_L. Since T_H can increase without affecting the midpoint temperature, the heat flow must be independent of TH. From the continuity of heat flow, we have:

(15)q=-\kappa \left(x,T\right){\displaystyle \frac{\mathrm{d}T}{\mathrm{d}x}}\equiv C,

where q is the heat flow density and C is a constant.

Applying the same formula to the scenario of changing T_L yields a similar result, demonstrating that the model is centrosymmetric. According to the equation, when T\leftarrow T_C, \mathrm{d}T/\mathrm{d}x\leftarrow 0. When the temperature gradient is large, \kappa \left(x,T\right) approaches infinity. Achieving this condition is impractical with current materials, so approximations are required.

The researchers proposed a one-dimensional thermal conductive model, analogous to a macroscopic diode, utilizing three components to form a thermal metamaterial device, as illustrated in Fig.18. Here, Type-A and Type-B are nonlinear materials with temperature-dependent thermal conductivities, following the logistic functions L\left(T\right), commonly used in logistic regressions. The central region comprises a conventional thermal material with high thermal conductivity {\kappa }_{\eta }.

In the theoretical analysis, Type-A functions as a good thermal conductor at high temperatures and an insulator at low temperatures, while Type-B exhibits the opposite behavior. For a given transition temperature T_C, the following relations hold:

(16)\begin{array}{rl}& {\kappa }_A=L\left(T_C-T\right)=\delta +\frac{\epsilon {\mathrm{e}}^{T-T_C}}{1+{\mathrm{e}}^{T-T_C}},\\ & {\kappa }_B=L\left(T-T_C\right)=\delta +\frac{\epsilon }{1+{\mathrm{e}}^{T-T_C}},\end{array}

where \delta is small and \epsilon is sufficiently large.

Applying Fourier’s law, geometrical considerations, and continuity of heat flow, the following relationship is derived (assuming {\mathrm{e}}^{T_0-T_C} and {\mathrm{e}}^{T_C-T_3} are close to zero):

(17)\begin{array}{rl}& \left(\delta +\epsilon \right)\left(T_3-T_2\right)-\epsilon \left[T_3-T_C-\delta \left(T_1-T_0\right)\right]\\ & =\epsilon \ln \left(\frac{1+{\mathrm{e}}^{T_1-T_C}}{1+{\mathrm{e}}^{T_2-T_C}}\right).\end{array}

To achieve the desired thermal stability, the thermal conductivity of the metamaterials central region must be significantly high ({\kappa }_{\eta }\gg \epsilon), ensuring T_1\approx T_2. Consequently, T_1\approx T_2\approx T_C is consistent with the theoretical model. This principle guides the design of thermal metamaterials capable of maintaining a stable internal temperature without energy input under varying external conditions.

The authors also suggested using shape memory alloys (SMAs) for thermal metamaterial design. SMAs exhibit temperature-dependent deformation, fitting the phase transition at T_C to achieve the required properties. In their design, Type-A and Type-B were configured using SMA materials with identical transition temperatures but opposite deformation directions. As the temperature of the heat or cold source changes, a bimetallic strip moves up or down, adjusting thermal conductivity as shown in Fig.19. Experimental results in Fig.19(B) showed that the temperature in the central region remained stable under various ambient temperatures, while it fluctuated significantly without the metamaterial in control experiments.

7.3.3 Thermal cloak-concentrator

By developing a theory of transformation heat transfer for multifunctional applications, this work introduces the concept of intelligent thermal metamaterials with dual functions [456]. The device can autonomously transition from a cloak to a concentrator, or vice versa, in response to external temperature variations.

Traditional thermal cloaks prevent heat transfer within the central region while maintaining the external temperature distribution, effectively hiding the object inside. As illustrated in Fig.20, a two-dimensional thermal cloak is designed using a compression transformation that maps a circle of radius R_2 to an annulus with inner radius R_1 and outer radius R_2. Conventional thermal concentrators channel more heat flux into the central region, thereby increasing the internal temperature gradient. The corresponding transformation for the thermal concentrator maps an annular region with an inner radius of R_3 and outer radius of R_2 to one with an inner radius of R_1 and outer radius of R_2.

Based on temperature-dependent nonlinear transformation thermotics, a bi-functional metamaterial is designed to switch between cloaking and concentrating functions based on the temperature T, see Fig.21. Theoretical analysis is used to elucidate this functionality. In a polar coordinate system, the thermal conductivity \kappa \left(T\right) of this bi-functional device is expressed as

(18)\begin{array}{rl}& {\kappa }_r\left(T\right)={\kappa }_0\left[1+\frac{R_2\left(R\left(T\right)-R_1\right)}{r^{\mathrm{\prime }}\left(R_2-R\left(T\right)\right)}\right],\\ & {\kappa }_{\theta }\left(T\right)={\kappa }_0{\left[1+\frac{R_2\left(R\left(T\right)-R_1\right)}{r^{\mathrm{\prime }}\left(R_2-R\left(T\right)\right)}\right]}^{-1},\end{array}

where {\kappa }_r\left(T\right) and {\kappa }_{\theta }\left(T\right) are the radial and azimuthal components of \kappa \left(T\right), respectively, and {\kappa }_0 is the background conductivity. For further details on the derivation of Eq. (18), refer to Ref. [456].

The function \kappa \left(T\right) is determined based on the desired functionality of the cloak-concentrator. The system is symmetric, resulting in two forms of transformation. These forms of \kappa \left(T\right) are

(19)\begin{array}{rl}& R\left(T\right)=\frac{R_3}{1+{\mathrm{e}}^{\beta \left(T-T_C\right)}}(\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{I}),\\ & R\left(T\right)=\frac{R_3{\mathrm{e}}^{\beta \left(T-T_C\right)}}{1+{\mathrm{e}}^{\beta \left(T-T_C\right)}}(\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{I}\mathrm{I}),\end{array}

where \beta is a scaling factor that defines the device’s sensitivity to temperature.

In Form I, the device acts as a concentrator when the ambient temperature is below the critical value T_C, directing more heat flux into the central region. As the ambient temperature exceeds T_C, the device functions as a thermal cloak, preventing heat flux from reaching the central region. Conversely, in Form II, the device operates as a thermal cloak at low temperatures and transitions to a concentrator at high temperatures.

Next, a bilayer concentrator is proposed, having the same configuration and dimensions as the cloak, with conductivities of regions 1−4 given by {\kappa }_1^{\mathrm{\prime }}, {\kappa }_2^{\mathrm{\prime }}, {\kappa }_3^{\mathrm{\prime }}, and {\kappa }_4^{\mathrm{\prime }}, respectively. To mitigate anisotropy introduced by transformation, a composite material is introduced in region 3, based on effective medium theory [457, 458]. The effective conductivity is given by {\kappa }_3^{\mathrm{\prime }}= \sqrt{{\kappa }_A{\kappa }_B}, where {\kappa }_A and {\kappa }_B are the conductivities of the two component materials. Region 3 is composed of perforated copper sheets filled with polydimethylsiloxane to achieve the desired thermal properties.

To meet the thermal concentration requirement, it is necessary to ensure that the thermal field in region 4 is undisturbed, while the temperature gradient in region 1 remains zero. To eliminate distortions induced by regions 24, the conductivity of region 4, {\kappa }_4^{\mathrm{\prime }}, should satisfy

(20){\kappa }_4^{\mathrm{\prime }}={\kappa }_3^{\mathrm{\prime }}{\displaystyle \frac{-b^2\left(b^2{l}_1+c^2{l}_2-a^2{\kappa }_3^{\mathrm{\prime }}\right)\left(c^2+b^2{l}_1{l}_2\right)}{ab\left(b^2{l}_1+c^2{l}_2-a^2\right)\left(c^2+b^2{l}_1{l}_2\right)}},

where l_1=\left({\kappa }_1^{\mathrm{\prime }}+{\kappa }_2^{\mathrm{\prime }}\right)/\left({\kappa }_1^{\mathrm{\prime }}-{\kappa }_2^{\mathrm{\prime }}\right) and l_2=\left({\kappa }_2^{\mathrm{\prime }}+{\kappa }_3^{\mathrm{\prime }}\right) /\left({\kappa }_2^{\mathrm{\prime }}-{\kappa }_3^{\mathrm{\prime }}\right). The above equation is a solution to Laplace’s equation, taking into account diffusion continuity.

As described in Eq. (18), R\left(T\right) represents a temperaturedependent function, implying temperature-dependent geometric deformation. In this theory, R\left(T\right) varies between 0 and R_3. With a sufficiently large \beta in Eq. (19), the exponential form of R^{\ast }\left(T\right) can be approximated as a step function with two distinct states. Similarly, in the bilayer design, the automatic transition between cloaking and concentrating can be achieved by adjusting the material in region 3.

To enable material responsiveness in region 3, a special bimetal strip is employed experimentally. The bimetal strip is fabricated by bonding a bidirectional SMA onto a metal sheet. Due to solid−solid phase transitions, these SMAs change shape at different temperatures, thereby facilitating the functionality of the component. In Form I, at temperatures of 297.2 K or higher, the SMA tilts upward to its maximum angle, thermally insulating region 3 and forming a thermal cloak. At lower temperatures, the SMA flattens to cover the insulator, creating a thermal concentrator. In Form II, the SMA deforms in the opposite manner.

7.4 Summary

This chapter comprehensively discusses temperature dependent transformation thermotics, covering both theoretical foundations and practical applications. Building upon traditional transformation thermodynamics, it introduces components designed for metamaterial applications based on temperature variations in nonlinear materials. Nonlinear transformation thermodynamics theory facilitates the design of multifunctional devices, enabling automatic transitions between functionalities through temperature changes. Based on this theory, functional transformation devices have been realized using isotropic materials and shape memory alloys, achieving capabilities unattainable by traditional thermal metamaterials. Given its applicability across various fields of physics, this theory holds significant potential for broad application and development.It is hoped that this work will inspire innovative ideas and designs in other disciplines.

8 Macroscopic thermal rectification: Temperature dependent parameters

Yuguang Qiu, Jiping Huang^*

　 Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China

8.1 Background

Designing metamaterials using the nonlinear properties of physical systems has already achieved great success in the electromagnetic field [459]. However, for thermal metamaterials, this approach is still in its early stages of development, especially macroscopic systems. The mechanisms for achieving nonlinear properties are diverse, with the two most common being the temperature dependence of material parameters and the size effects in microscale systems [460]. However, macroscopic heat conduction lacks the size-dependent mechanisms found in microscopic systems. This means that while heat transfer processes at the microscale are typically nonlinear, the macroscopic Fourier heat transfer equation exhibits a linear response between heat flux and temperature gradient. Therefore, macroscopic nonlinearity specifically refers to the temperature dependence of parameters, especially thermal conductivity, in the Fourier equation. Investigating nonlinear behavior in macroscopic heat transfer systems and developing metamaterial designs distinct from linear regimes represents a highly promising research direction.

The thermal diode, or thermal rectifier, is the most typical example of nonlinear thermal metamaterial design [461–464]. Its asymmetric heat transfer capability makes them significantly important in areas such as energy harvesting [465] and the design of thermal logic gates [466]. Fig.22 outlines the framework for achieving thermal rectification using nonlinear effects and presents the future outlook for nonlinear thermodynamics and thermal metamaterials. As early as the 1970s, experiments demonstrated that thermal rectification could be achieved by connecting two materials with different temperature responses [467], as shown in Fig.23. This structure is straightforward to simulate and experiment with, making it the most commonly used structure for rectification to this day. However, broader research into nonlinear properties inducing thermal rectification has only gradually emerged over the past two decades. And initially, this research was mostly limited to the microscopic level, with studies on Fourier heat transfer becoming popular only in the last decade. The following sections will briefly summarize the development of this field from both theoretical and application perspectives, and provide an outlook on future directions.

8.2 Past to current development

8.2.1 Theory

The development of theory in this field can be summarized in two parts: first, exploring how the nonlinearity of parameters, particularly thermal conductivity, determines the properties of rectification systems; and second, exploring various physical mechanisms that can achieve nonlinearity.

How nonlinearity generates thermal rectification is a primary question for researchers. Simply having a thermal conductivity that depends on temperature is not sufficient for rectification. In a system with a symmetric structure, a necessary condition for achieving thermal rectification is that the thermal conductivity must be a function of both temperature and spatial position, with these two parameters being inseparable [468, 469]. However, this condition alone is not sufficient [469]. When the system’s structure is asymmetric in the direction of heat transfer, thermal rectification can be achieved regardless of whether the parameters are separable [470].

One major criterion for assessing the usefulness of rectification devices is the degree of rectification they achieve. So, once rectification behavior is established, the next core consideration is how to enhance the rectification ratio. One simple method is to use an asymmetric geometric structure [471, 472]. However, since the material’s nonlinearity arises from the temperature difference, a more significant change in thermal conductivity requires a larger temperature difference. Therefore, the maximum rectification ratio also depends on the system’s temperature difference and the material’s nonlinear response. So, even with an asymmetric structure, the rectification ratio remains low when the temperature difference is small. The introduction of phase change materials has enabled the achievement of high rectification ratios even at low temperature differences [473]. These materials exhibit a thermal conductivity response to temperature that resembles a step function, with distinctly different conductivities above and below a phase transition temperature. A sudden change in parameters represents a significant nonlinear property, which can achieve extremely high rectification ratios and even function as a thermal diode near the phase transition temperature.

With a theoretical framework for achieving rectification in place, the next focus is on how to specifically induce nonlinearity, particularly abrupt changes in parameters. One approach is to search for suitable nonlinear materials, such as specific chemical compounds or artificial alloys that exhibit temperature-dependent properties [474]. The phase transition of a material can result in corresponding changes in thermal conductivity with temperature, such as the metal-insulator transition in {\mathrm{V}\mathrm{O}}_2 [475]. A simple example is the use of a material’s solid−liquid phase transition to achieve changes in thermal conductivity; even ordinary {\mathrm{H}}_2\mathrm{O} can exhibit this property [476]. Another mechanism to achieve sudden changes in thermal conductivity is through mechanical structures that create moving contact interfaces. For example, shape memory alloys can exhibit phase change material-like properties. When deformed at low temperatures, a shape memory alloy will return to its original shape once the temperature exceeds a certain activating temperature. This characteristic allows shape memory alloy to effectively emulate phase change material behavior [477]. At low temperatures, bending the alloy reduces contact between materials, thereby decreasing heat transfer. When the temperature rises and the alloy returns to its original shape, the contact area increases, and the heat transfer correspondingly improves.

While the use of nonlinear materials is straightforward, their temperature response functions are often fixed. If the intensity of the nonlinear response could be continuously adjusted, it would greatly expand the range of applications for metamaterial designs. By applying effective medium theory, uniformly doping one nonlinear material with another material that has a different response intensity can produce an enhanced or diminished nonlinear coefficient. By changing the doping ratio, continuous control of the nonlinear coefficient can be achieved [478].

Moreover, combining multiple physical fields is another approach to achieving nonlinear heat transfer. According to the Stefan−Boltzmann law, the relationship between radiative heat flux and temperature is inherently nonlinear. If heat radiation is combined with heat conduction, a nonlinear response between heat flow and temperature gradient can be obtained naturally. Particularly in far-field situations, the Rosseland diffusion model shows that radiative heat flux is also proportional to the temperature gradient, making it a reasonable additional nonlinear term in the Fourier conduction equation [479]. Additionally, the temperature dependence of emissivity ϵ(T) can also produce rectification effects [480]. A similar mechanism appears in convective heat transfer. For instance, when the medium is a ferromagnetic fluid, thermal rectification can be achieved through the temperature-dependent magnetization M(T) [481]. Besides coupling the forms of heat transfer, multiphysics also manifests in coupling thermal properties with other physical properties, such as using electric fields or magnetic fields to control phonon heat transfer to achieve nonlinearity [480], or introducing mechanical mechanisms [482]. These multiphysics couplings, can also combined with different heat transfer modes, imply more complex systems capable of achieving richer properties. Moreover, through machine learning, the limitation of narrow applicable temperature range can be further overcome [483].

8.2.2 Application

While theoretical advancements have progressed, thermal rectifier devices have also spawned a variety of practical applications.

Energy regulation, including insulation, storage, conversion, and recovery, represents the primary application area for thermal rectification. The asymmetric heat transfer in thermal rectifiers enables unidirectional cooling and waste heat scavenging between two regions [480, 484, 485]. And the use of phase-change materials further enriches the design possibilities for thermal rectification. This abrupt response allows temperature changes to act like a switch that alters thermal conductivity. Using this temperature switch, not only can thermal diodes be achieved, but systems can also switch between different states at high and low temperatures. A typical example is combining phase change materials with transformation theory [477] or scattering cancellation theory [486] to switch between normal and thermal stealth states. Based on this macroscopic thermal diode design, researchers have developed an energy-free thermostat technology. By combining two phase change materials with opposite switching behaviors and a high thermal conductivity material into a sandwich structure, the temperature in the high thermal conductivity middle layer remains constant despite significant environmental temperature fluctuations, without requiring any additional energy [487]. Furthermore, because the temperature difference is concentrated in the phase change materials on either side, a thermal voltage can be generated to recover electrical energy, achieving negative energy consumption while maintaining constant temperature [488]. Additionally, due to the characteristics of thermal rectification, temperature differences of equal magnitude but opposite directions produce varying degrees of heat flow. This means that under dynamically changing heat sources, heat transfer can occur even with a zero or negative average temperature gradient [489], which can facilitate waste energy recovery.

Another direct application idea is to draw an analogy with electronic diodes. Electronic diodes are now ubiquitous fundamental components in the semiconductor field. Similarly, thermal diodes can be used to implement basic logic gates such as AND, OR, and NOT [466]. And heterostructures composed of materials with different temperature responses can achieve bistable or even multistable systems, allowing for the storage of larger amounts of thermal information [490, 491]. With these analogies to thermal diodes and thermal switches, many applications from electrical circuits can be translated to the thermal domain [480].

8.3 Future outlook

Although the exploration of thermal circuits has yielded some preliminary results, the inherent dissipative nature of diffusive systems makes it challenging to achieve more dense and complex applications. Looking ahead, beyond further research into realizing nonlinearity and improving rectification ratios, integrating thermal diodes for communication [492] using thermal signals would mark a significant leap in the application of nonlinear thermal devices.

The expansion of applications also demands higher material properties. On one hand, some existing materials like shape memory alloys still suffer from limitations such as instability and operational difficulties. On the other hand, designing intelligent devices is a new challenge in the field of metamaterials. Features like automatic environmental regulation and multifunctional switching can better meet diverse application needs [477]. Stabilizing, simplifying and enhancing the intelligence of materials will be a key direction for the future.

Taking a broader perspective on the entire field of nonlinear thermal physics, it may also be valuable to consider its analogy to nonlinear optics. Nonlinear optics is a well-established field, with its nonlinear theories primarily based on the dependence of the dielectric constant on the electric field intensity, which is quite similar to the dependence of thermal conductivity on temperature. Currently, higher harmonics generation, akin to that in optics, has been achieved with sinusoidal heat sources in nonlinear materials [478]. Such work is referred to as nonlinear thermotics [474, 477–479].

However, there are fundamental differences between the mathematical version of thermal systems, which are diffusive, and optical systems, which are wave-based. Additionally, in optics, the nonlinear term is the coefficient of the second spatial derivative of the field function, {\mathrm{\partial }}^{2}D/\mathrm{\partial }{t}^2= {ϵ}_0{c}^2{\mathrm{\nabla }}^{2}E, whereas in the heat diffusion equation, the nonlinear term is the coefficient of the first spatial derivative, \rho C\mathrm{\partial } T/ \mathrm{\partial }t= \mathrm{\nabla }\cdot \left[k(T)\mathrm{\nabla }T\right]. Furthermore, the dielectric constant can be negative or have an imaginary component, while such forms lack a physical basis in thermal conductivity. These differences suggest that expanding the principles of nonlinear optics into the realm of nonlinear thermal systems may lead to entirely new physical phenomena. This has been preliminarily explored in the field of non-Fourier heat transfer [493], and how to realize analogy in macroscopic Fourier heat transfer is a potential direction for future research.

8.4 Summary

Incorporating the temperature dependence of parameters into traditional thermal metamaterials opens new avenues for thermal regulation design. Theoretically, introducing nonlinearity breaks the symmetry of heat transfer capabilities between high and low temperatures, making it a natural choice for achieving thermal rectification. This foundation has led to the development of a series of more sophisticated thermal regulation devices. Experimentally, the focus lies on achieving ideal nonlinear responses and further integrating these applications. It is important to note that while these theories can be applied at the microscopic level, our emphasis here is on macroscopic regulation, particularly extending the Fourier equation, this also facilitates comparisons with nonlinear behaviors in other physical fields.

9 Non-Fourier steady-state and high-frequency heat conduction in thermal metamaterials: A space-time discrete model

S. L. Sobolev^1,2*

　 ¹Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry, Russian Academy of Sciences, Chernogolovka Moscow Region 132432, Russia

　 ²Samara State Technical University, ul. Molodogvardeiskaya 244, Samara 443100, Russia

9.1 Background

Recent advancements in thermal metamaterials (TM) have offered a promising tool to control heat flow at both time and space scales, leading to exciting applications such as adjustable, reconfigurable, and intelligent thermal metadevices [494–500]. Coherent control of wave-like heat propagation phenomena using TM has been the driving force of the technological revolution in different technologically important fields ranging from electronics, photonics to phononics [500–506]. Thus, the heat flow in TM is an important area of study, not only with technological implications, but also with fundamental research. The problem is that the most studies of heat flow in TM are based on the Fourier law [494–500], which describes heat transport as a purely diffusive process and is applicable only to relatively slow processes on a long space scale. Recently, a few studies have considered transient heat conduction in TM using continuous generalizations of the Fourier law [501–506], namely hyperbolic heat equation and so-called dual-phase-lag equation. While the hyperbolic equation slightly extends the domain of applicability of the Fourier law [507–509], but still is not applicable to high frequency (fast time scale) heat conduction in a finite (nano) domain, the dual-phase-lag equation has no any reasonable physical basis at all. Note that any modifications of the Fourier law in the form of the continuous partial differential equations (PDEs) require knowledge of additional boundary and initial conditions that cannot be physically determined [510]. Moreover, the continuum description based on the PDFs in dielectrics materials, for example in graphene-like materials, where thermal transport is not localized in space because the frequencies and wave numbers in a crystal are discrete and limited, is not applicable at all [511].

In contrast to the conventional continuous description in the form of the PDE, the discrete variable model (DVM) assumes that the space and time are discrete variables, and provides evolution equations for temperature and heat flow in the discrete from [510, 512–518]. The DVM takes into account the inherent space-time nonlocality of the transfer processes and is applicable on ultrashort space and time scales. While the lattice model assumes that time is a continuous variable [519], the DVM suggests that both time and space are discrete. The main purpose of this short review is to provide a general view of present achievements of the discrete approach from both theoretical and application perspectives, paying special attention to high-frequency heat transport in the layered TM.

9.2 Past to current development

9.2.1 Theory

A wide class of TM are represented by the layered metamaterials [494–499, 504, 505] produced by the layer-upon-layer joining of the material such as in the 3D printing and additive manufacturing or the laser-based directed energy deposition [520]. A periodic 1D structure composed of layers of two different materials indexed A and B is schematically shown in Fig.24(a). This configuration corresponds, for example, to temperonic crystals [504] and acoustic metamaterials [521]. The thicknesses of the layers are denoted as h_A and h_B, respectively. Each discrete layer indexed by j is characterized by heat capacity C_j= c_j{\rho }_j{h}_j, where h_j is the length of the layer, c_j is the specific heat, {\rho }_j is mass density, i.e., thermal properties of the system change periodically in space [see Fig.24(a)].

In such a case the temperature inside a discrete layer changes insignificantly in comparison with the temperature jumps between the layers. Thus, it is assumed that the temperature of a layer j, denoted T_{j,n}, is constant inside the layer and evolves at the discrete time instants n=0,1 ,2,\dots due to the heat fluxes across the boundaries of the layers (see Fig.24). The heat flux between the layers j-1 and j, denoted q_{j-1,j}, is proportional to the boundary temperature jump, i.e., the temperature difference between the layers, and is given by q_{j-1,j}\sim \alpha \left(T_{j-1,n}-T_{j,n}\right), where \alpha is the heat exchange coefficient between the neighboring layers (the inverse of the heat exchange coefficients R=1/\alpha is also referred to as thermal resistance). This formulation takes into account that the temperature is a volume property, whereas the heat flux is a boundary (surface) property (see Fig.24). The discrete heat equation (DE) for the temperature takes the form [517, 518]

(21)\left(T_{j,n+1}-T_{j,n}\right)={\displaystyle \frac{{\beta }_j}{2}} \left(T_{j-1,n}-2T_{j,n}+T_{j+1,n}\right),

where {\beta }_j=2\alpha /C_j. In terms of the continuous variables x and t that are related with the discrete ones as follows x=jh and t=n\tau. In such a case, Eq. (21) is given by

(22)\begin{array}{rl}T(x,t+\tau )-T(x)=& \frac{\beta (x)}{2}[T\left(x-\frac{h_A+h_B}{2},t\right)-2T(x,t)\\ & +T\left(x+\frac{h_A+h_B}{2},t\right)],\end{array}

here \beta (x)=2\alpha /C(x), where C(x) is given by

(23)

Eqs. (21−23) can be used directly for computer simulations, but in what follows, we are going to slightly simplify Eq. (22) to be able to provide physical insights into the problem without much of numerical efforts. We assume that h_A\gg h_B [see Fig.24(b)]. In such a case, C_A\gg C_B and, consequently, the layers B do not affect the energy balance in the system, but strongly affect the temperature distribution resulting in the thermal resistance between the layers A. Manipulating the physical properties of the layer B and/or its thickness, one can change the value of the interfacial resistance and govern heat conduction regimes in the system. Thus, we consider a periodic one-layer system consisting of identical layers A of the same length h [see Fig.24(b)] and with the same heat capacity C_j= C, whereas the heat exchange coefficient between layers A depends now on the set of thermal properties of the layer B. In such a case, Eq. (22) yields [517, 518]

(24)T(x,t+\tau )-T(x,t)={\displaystyle \frac{\beta }{2}}[T(x+h,t)-2T(x,t)+T(x-h,t)],

or in operator form

(25)\left[\exp \left(\tau {\mathrm{\partial }}_t\right)-\beta \cosh \left(h{\mathrm{\partial }}_x\right)\right]T(t, x)=1 -\beta .

To bridge the gap between discrete and continuum approaches, we consider discrete Eq. (24) in the continuum limit \tau \to 0 and h\to 0. The Taylor of the DE, Eq. (24), around \tau =0 and h=0 contains an infinite number of terms with two small parameters \tau and h. To obtain continuum partial differential equations (PDEs) with a finite number of terms, it is necessary to balance the fast-time \tau and the short-space h scales [513–515].

Under assumption that D=h^2/2\tau preserves a finite value when \tau \to 0 and h\to 0, the first-order approximation of Eq. (27) leads to the classical Fourier equation of parabolic type \mathrm{\partial }T/\mathrm{\partial }t=\beta D{\mathrm{\partial }}^{2}T /\mathrm{\partial }{x}^2 [513–515]. The second-order approximation gives \mathrm{\partial } T/ \mathrm{\partial }t+ (\tau /2){\mathrm{\partial }}^{2}T/\mathrm{\partial }{t}^2= \beta D{\mathrm{\partial }}^{2}T/\mathrm{\partial }{x}^2+\left(\beta D{h}^2/24\right){\mathrm{\partial }}^{4}T/\mathrm{\partial }{x}^2. Taking into account that {\mathrm{\partial }}^4/\mathrm{\partial }{x}^4= (1/\beta D){\mathrm{\partial }}^3/\mathrm{\partial }t\mathrm{\partial }{x}^2, the last equation transforms into the 1D Guyer−Krumhansl or Jeffreys type equations [507]. These equations, as well as higher order approximations, are the PDE of parabolic type with an infinite velocity of propagation of temperature disturbances v=h/\tau. Indeed, v=(D /h )\to \mathrm{\infty } when D preserves a finite value at \tau \to 0 and h\to 0. The second invariant of the continualization procedure keeps a finite value of propagation velocity when \tau \to 0 and h\to 0 [513–515]. In such a case, the first-order approximation of Eq. (27) leads to well-known hyperbolic heat (telegraph) equation \mathrm{\partial }T/\mathrm{\partial }t+( \tau /2){\mathrm{\partial }}^{2}T/\mathrm{\partial }{t}^2=(\beta v^2\tau /2){\mathrm{\partial }}^{2}T/\mathrm{\partial }{x}^2. Similar to the previous case, the effective thermal diffusivity given by \beta v^2\tau /2 depends on the heat exchange coefficient \beta (or thermal resistance 1/\beta). According to the second invariant of the continualization procedure, this and higher order approximations are of hyperbolic type with a finite velocity of temperature disturbances. Thus, DE, Eq. (24), contains an infinite hierarchy of continuous PDEs of both hyperbolic and parabolic types including the classical Fourier law, hyperbolic heat equation, and GK or Jeffreys type equation. These PDEs are truncated Taylor series expansions of the DE, Eq. (24), and approximate it with some accuracy depending of the number of terms taken into account and the invariant of the continualization procedure. The model chosen to describe a particular heat conduction process depends on what level of accuracy is required in the analysis and what mode of heat transport (diffusive or wave) plays the most important role. Generally, the DE equation is superior for simulations of heat transport in metamaterials on ultrashort space-time scales.

To study the dispersion properties of the DE, we use the standard procedure seeking for solution of Eq. (4) in the form T=T_0\exp [ \mathrm{i}(\omega t+k x)], where \omega is the real frequency and k is the complex wave number. In such a case, the dispersion relation for Eq. (25) is given by [518]

(26)\exp (\mathrm{i}\omega \tau )- \beta \cos (kh) =1-\beta .

When \beta =1, Eq. (21) reduces to T_{n+1,j}=(T_{n,j+1}+T_{n,j-1})/2 and Eq. (26) gives \exp \left(\mathrm{i}\omega \tau \right)=\cos \left(kh\right) [517].

Fig.25 shows the normalized real Rekh and imaginary Imkh parts of the wave number k, as well as the phase and group velocities, as functions of normalized frequency \omega \tau in the range 0\le \omega \tau \le \pi obtained from Eq. (26) [517, 518]. These functions demonstrate unusual behavior at high frequency compared to the continuum case [517, 518]. The maximum allowed frequency and wave vector are restricted (there are no such limits in a continuum description). The group velocity may exceed the phase velocity, whereas both velocities may exceed velocity of propagation of temperature disturbances (see Fig.25). When \omega \tau \to \pi, the group velocity tends to zero at \beta = 1, whereas the phase velocity tends to velocity of thermal disturbances v at any value of \beta (for more details see Refs. [517, 518]). Penetration depth (or attenuation distance) defined as d=1/\mathrm{I}\mathrm{m}k in case of the Fourier law monotonically decreases from infinity at \omega \tau \to 0 to zero at \omega \tau \to \mathrm{\infty } [517, 518]. The penetration depth for the DE, Eq. (24), defined in the same way as d_{DE} =1/\mathrm{I}\mathrm{m}(k), \mathrm{a}\mathrm{t}0\le \omega \tau \le \pi /2, behaves in a similar way as the penetration depth for Fourier law d_F, i.e., tends to infinity and decreases to zero with increasing \omega \tau. However, at \pi /2\le \omega \tau \le \pi, d_{DE} demonstrates an opposite behavior, i.e., it increases and tends to infinity at \omega \tau \to \pi [517, 518]. The penetration depth can be defined in another way. Indeed, a sudden jump of temperature on the boundary x>0 of the 1D dynamic TM propagates into the bulk x>0 with constant velocity v=h/\tau due to the discrete structure of DE, Eq. (24) [513, 517, 518]. This fast wave is followed by relatively slow diffusive propagation of heat with the Fourier penetration depth d_F={\left(\beta Dt\right)}^{1/2} [508].

9.2.2 Applications

The discrete description is relevant for processes of many physical contexts, such as calcium burst waves in living cells, propagation of action potentials through the tissue of the cardiac cells, chains of neurons or chemical reactions, the local denaturation of the DNA double strand [522]. The two-dimensional (2D) version of the DVM [513–515] has been used to study information conduction and convection in noiseless Vicsek flocks [523]. The discrete approach is also applicable to high frequency vibrations, behavior of material near cracks and fronts of destruction waves [516] and to many other practically important situations. In particular, the DVM can be used to study heat transport in the so-called static TM, which have a wide range of potential applications, including thermal cloaking, energy harvesting, thermoelectric devices, and so on [494]. For example, the static temperature field is observed in steady-state heat conduction across the layered TM placed between two thermal baths (see Fig.26). In this case, the DVM leads to analytical expressions for the effective thermal conductivity, boundary temperature jumps, and heat flow across the TM as functions of the number of the discrete layers N and thermal resistance between the layers [525]. In addition, the DVM demonstrates that the local thermal conductivity in the layered TM is position dependent and introduces thermal extrapolation length, which virtually eliminates the temperature jump at the boundaries with thermal baths and makes molecular dynamic simulations more effective [524]. Note that these effects are not predicted by the continuum Fourier-like approach. Moreover, the DVM can be used to introduce effective temperature for the local nonequilibrium state [526, 527].

9.3 Future outlook

The present DVM is rather universal and applicable at any level of description: microscopic, mesoscopic or macroscopic, and to both static and dynamic transport in TM, as long as the variations of temperature inside the discrete cells or layers are relatively small in comparison with the temperature jumps between the layers. The DVM takes into account both diffusive and wave (ballistic-like) modes of transport processes which makes it a promising candidate for the description of temperature waves in TM on short space and time scales. Note that the continuous Fourier approach takes into account only the diffusive mode and cannot be used to study wave-like heat conduction in TM.

On microscopic level, the discretization space scale h corresponds to a lattice constant for a crystal and is of the order of an atomic-level length scale [511]. In the context of heat transport on the micro scale, h may be associated with the mean free path of energy carriers, which covers a wide range from a few nanometers to a few microns, while \tau - with the mean free time [511, 527]. On this level, the DVM can be used to study heat transport in the lattice structures, such as layered correlated materials [505] or graphene-like materials [506], as well the thermal transport from 1D- and 2D-confined nanoscale hot spots (periodic nanolines and nanodots) [528].

On mesoscopic level, the DVM is applicable to heat conduction in the TM which thermal properties change periodically [see Fig.24(a)]. In particular, a wide class of TM is represented by the layered metamaterials produced by the layer-upon-layer joining of the material such as in the 3D printing and additive manufacturing or in the laser-based directed energy deposition [520, 521]. For these metamaterials h corresponds to the thickness of the layers (see Fig.24 and Fig.26). A temperonic crystal (TC), which represents a periodic structure with a unit cell made of two slabs sustaining temperature wavelike oscillations on short timescales, also serves as an example of such a mesoscale system [504]. Note that the TC is the analog, for the temperature case, of electronic, phononic, and photonic superlattices [504]. Other examples of possible application of the DVM are provided by (i) acoustic metamaterials with unit cells composed of two layers made of dissimilar materials with a crack-like void situated at the interface between bars [521]; (ii) multi-layered films for active food packaging applications, which show better mechanical, barrier and controlled release properties compared to monolayered films [529]; (iii) various renewable energy and solar-thermal technologies of important practical applications, such as solar thermionic, solar thermoelectric, and concentrated solar power where the nanostructured multilayer metamaterials play a key role [530]; (iv) layered metamaterials for quantum technologies, including quantum light sources, photon detectors and nanoscale sensors [511]. The 2D version of DVM [513–515] adds a new tool for the investigation of both static and high-frequency heat transport in graphene and other technologically relevant 2D TM. The two-temperature generalization of the DVM in 2D [514] is applicable to study heat transport in cellular TM represented by the two-phase composite media schematically shown in Fig.27. Typical examples are nanogranual or cellular TM, which in perspective may be good candidates for seeking wave-like heat conduction regimes [501, 503]. If the different phases in the composite media have different thermal parameters, this may lead to the temperature differences between the phases during dynamic high-frequency processes. In such a case, the phases can be described by their own temperatures, T_1 and T_2, as for example in the continuum two-temperature models [532–534] or in mass-in-mass chains [535]. 2D heat conduction in the two-phase TM (see Fig.27) is governed by two coupled discrete equations with allowance for energy exchange between the cells of both phases [514]. Further generalization of the DVM may include: (i) phonon dispersion on micro level; (ii) more detailed description of the 2D [514] and 3D cases with different geometries, (iii) multi-temperature DVM for multi-phase TM - similar to the multi-temperature model of heat transfer in solids under ultrashort laser irradiation [534]; (iv) application of the DVM, with appropriate modifications, to mass transport problems [535]; (v) semi-continuum limit, when the time is continuous but the space is discrete [518, 519, 536] (or vice versa).

9.4 Summary

The DVM takes into account the inherent space-time nonlocality of the transfer processes and, is applicable to the study of both static and dynamic heat conduction in TM at any level of description (microscopic, mesoscopic or macroscopic). The DVM predicts rather unusual behavior of heat transport in the multi-layered TM, particular for the high-frequency and wave-like regimes [517, 518]. This unusual behavior opens up new perspectives in the development of the multi-layered TM and may extend a range of possible technological applications. It is hoped that this short review will stimulate future theoretical and experimental investigations of heat transport in TM on ultrashort space and time scales.

10 Directional heat transfer: Extremely anisotropic parameters

Wenrui Liao, Fubao Yang, Liujun Xu^*

　 Graduate School of China Academy of Engineering Physics, Beijing 100193, China

10.1 Background

Designing metamaterials with transformation theory has achieved great success in the thermal field [537–541]. However, the dependence of traditional metamaterials on background parameters and device shapes makes them apply to only certain scenarios. A new type of metamaterials with extremely anisotropic parameters was proposed to overcome these disadvantages. Extreme anisotropy means the thermal conductivity in one direction approaches infinity while being zero in the other(s), exhibiting unexpected properties, such as chameleonlike behaviors [542] and transformation-invariant features [543, 544]. Achieving extremely anisotropic parameters is challenging with natural bulk materials. Thus, a multilayered composite structure is designed by alternately arranging two isotropic materials, whose effective thermal conductivity exhibits high anisotropy.

A single-layer thermal cloak is the most typical design of thermal metamaterials [537]. According to transformation theory, the transformed thermal conductivity should satisfy {\kappa }_{rr}^{\prime }={\kappa }_0(r^{\prime }-a)/r^{\prime } and {\kappa }_{\theta \theta }^{\prime }={\kappa }_0{r}^{\prime }/(r^{\prime }-a), where {\kappa }_{rr}^{\prime }, {\kappa }_{\theta \theta }^{\prime }, {\kappa }_0, and a are the radial and angular thermal conductivity of the shell, background thermal conductivity, and the inner radius of the shell. The nonuniform thermal conductivity and its singularity at r^{\prime }=a make an ideal cloak difficult to realize in practice. Alternatively, we can replace the thermal conductivity with {\kappa }_{rr} =C and {\kappa }_{\theta \theta }=1/C (with a constant C) to maintain the cloaking effect [545]. The smaller C (i.e., the larger difference between {\kappa }_{rr}^{\prime } and {\kappa }_{\theta \theta }^{\prime }) generates better performance. Thus, extremely anisotropic parameters were introduced to thermal metamaterials. Besides, they have also emerged in various physical fields, such as electromagnetism [546, 547], electrostatics [548], thermotics [549–553], acoustics [554, 555], and mass diffusion [556]. In the following sections, we will introduce the development of extreme anisotropy in thermotics using three theoretical methods (i.e., transformation theory, scattering cancellation, and their combination). Since extremely anisotropic thermal conductivity confines heat flux to flow along a certain direction, directional heat transfer can be expected, which has broad applications in feasible and adaptive heat control. Finally, we provide an outlook for future development.

10.2 Past to current development

10.2.1 Transformation theory

A typical method to produce extremely anisotropic metamaterials is to stretch a medium with near-zero width [Fig.28(a)]. According to transformation theory, the thermal conductivity in the stretching direction (i.e., the main axis) will approach infinity, while the other(s) will become zero. For instance, we consider a 2D slab of near-zero width W_{\mathrm{v}} with isotropic thermal conductivity {\kappa }_0 and map it to a slab of width W_{\mathrm{p}}. The transformed thermal conductivity is {\mathit{\kappa }}^{\prime }={\kappa }_0\cdot \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( W_{\mathrm{p}}/W_{\mathrm{v}},W_{\mathrm{v}}/W_{\mathrm{p}}). Since W_{\mathrm{v}}\to 0 and W_{\mathrm{p}} is finite, the thermal conductivity finally becomes {\mathit{\kappa }}^{\prime }= {\kappa }_0\cdot \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{\infty },0). Such a thermal metamaterial is also called thermal null medium [552, 553].

Infinite thermal conductivity along the main axis guides the heat to transfer from the input surface to the output without any deviation, just like the space between the two surfaces does not exist. Each point on the input surface is mapped to a corresponding point on the output. Therefore, an important function of extremely anisotropic metamaterials is to realize directional heat transfer, which forms the foundation of various metamaterials with interesting functions [557, 558]. For instance, since the heat flows along the main axis, it can be deflected at will by connecting two extremely anisotropic metamaterials with an angle between the main axis [552, 553]. Besides, extremely anisotropic metamaterials can also achieve radiative thermal camouflage [559]. For this purpose, the background is covered with an extremely anisotropic metamaterial, making the temperature distribution of the background and that outside the surface of the metamaterial identical. Since thermal radiation depends on temperature, the same temperature profiles indicate that the object hidden inside the extremely anisotropic metamaterial is camouflaged.

Extremely anisotropic metamaterials can work with arbitrary shapes. See Fig.28(b), the material is divided into two regions by two curves R_2(\theta ) and R_3(\theta ) with the same shape factor R(\theta ). Then, the region is compressed into a smaller one , while the region r\in [R_2(\theta ),R_3(\theta )] is stretched into r^{\prime }\in [R_1 (\theta ),R_3 (\theta )]. Generally, the transformed thermal conductivity tensor in the stretched region has off-diagonal components. However, by imposing the null medium transformation, i.e., the thickness of the stretched region is near-zero, the diagonal components become {\kappa }_{rr}=1 /\mathrm{\Delta } and {\kappa }_{\theta \theta }=\mathrm{\Delta } with \mathrm{\Delta }\to 0. The steady-state heat conduction is governed by

(27)\mathrm{\nabla }\cdot (-\kappa \mathrm{\nabla }T)=0.

Since the off-diagonal components always include \mathrm{\Delta } by substituting the thermal conductivity into Eq. (27), the specific value of the off-diagonal components has little influence on heat transfer, which can thus be set as zero. The thermal conductivity eventually becomes \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{\infty },0). The heat flux is concentrated into the inner region owing to the compression transformation. According to transformation theory, the enhancement is R_2(\theta )/R_1(\theta ).

Therefore, extremely anisotropic parameters can be obtained from the null medium transformation to implement various functions like cloaking and concentrating.

10.2.2 Scattering cancellation theory

We consider a 2D core−shell structure to further investigate the properties of extremely anisotropic metamaterials [Fig.29(a)]. The thermal conductivity of the shell is {\mathit{\kappa }}_{\mathrm{s}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\kappa }_{rr},{\kappa }_{\theta \theta }), and that of the core and background are set as the same value {\kappa }_{\mathrm{c}}. The heat conduction equation [Eq. (27)] can be expressed in cylindrical coordinate system as

(28){\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }r}}\left(r{\kappa }_{rr}{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }r}}\right)+{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\theta }}\left({\displaystyle \frac{{\kappa }_{\theta \theta }}{r}}{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }\theta }}\right)=0.

Solving Eq. (28) with boundary conditions yields the temperature distribution in each region. We assume the undistorted external thermal field and derive the effective thermal conductivity {\kappa }_{\mathrm{e}} of the core−shell structure. In particular, substituting the extremely anisotropic parameters {\mathit{\kappa }}_{\mathrm{s}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( \mathrm{\infty },0 ) into the effective thermal conductivity yields {\kappa }_{\mathrm{e}}\approx {\kappa }_{\mathrm{c}}, which implies that the property of the core is extended to the whole structure. Note that {\kappa }_{\mathrm{e}}\approx {\kappa }_{\mathrm{c}} is always satisfied regardless of any other parameters. In other words, when {\kappa }_{\mathrm{c}} changes, {\kappa }_{\mathrm{e}} will spontaneously adapt itself to the value of {\kappa }_{\mathrm{c}}, maintaining the external field undistorted. In this regard, such a core−shell structure is termed chameleonlike [542, 549, 550]. The thermal conductivity \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{\infty },0) also indicates that it works as a concentrator.

Note that in 3D, {\kappa }_{\mathrm{e}}\approx {\kappa }_{\mathrm{c}} is no longer satisfied for extremely anisotropic parameters. Nevertheless, there still exists another types of chameleonlike metamaterials, which can be derived from solving the equation {\kappa }_{\mathrm{e}}={\kappa }_{\mathrm{c}}. According to the value of {\kappa }_{\theta \theta }/{\kappa }_{rr}, there are two solutions: when {\kappa }_{\theta \theta }/{\kappa }_{rr}>-1/8, the solution is {\kappa }_{\mathrm{c}}+{\kappa }_{rr} (1+\sqrt{1+8{\kappa }_{\theta \theta }/{\kappa }_{rr}})/2=0; when , the solution is (\sqrt{-1-8{\kappa }_{\theta \theta }/{\kappa }_{rr}}/2)\ln (R_{\mathrm{c}}/R_{\mathrm{s}})=-Z^{+}\pi (Z^{+}=1,2,3,\cdots ), where R_{\mathrm{c}} and R_{\mathrm{s}} are the radii of the core and the shell. The metashells with such anisotropic parameters still preserve chameleonlike behaviors in 3D (both the two solutions can be reduced to 2D) [542]. However, compared to extremely anisotropic metamaterials, these two types of metamaterials have some features: for the former, the thermal conductivity of the core and shell are restricted to a special relation; for the latter, when {\kappa }_{\mathrm{s}} is determined, the value of the core fraction R_{\mathrm{c}}/R_{\mathrm{s}} is correspondingly determined. In contrast, extremely anisotropic metamaterials does not require certain thermal conductivity and core fraction.

Therefore, we reveal the chameleonlike behaviors of extremely anisotropic metamaterials directly through scattering cancellation theory, which have potential applications in intelligent heat control.

10.2.3 Combining two theories

The extremely anisotropic metamaterials can preserve the eigenvalues of the thermal conductivity under an arbitrary coordinate transformation. We consider an arbitrary coordinate transformation r^{\prime }=R( r,\theta ) and {\theta }^{\prime }= \mathrm{\Theta }( r,\theta ). The transformed thermal conductivity is {\mathit{\kappa }}^{\prime }=\mathit{J}{\mathit{\kappa }}_0 {\mathit{J}}^{†}/det\mathit{J}, where \mathit{J} is the Jacobian matrix of the transformation, and the superscript † represents transpose. Generally, the components of the thermal conductivity tensor are determined by the specific form of the transformation. However, by imposing {\mathit{\kappa }}_0=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( \mathrm{\infty },0 ), the eigenvalues of {\mathit{\kappa }}^{\prime } are {\lambda }_1\approx \mathrm{\infty } and {\lambda }_2\approx 0. Thus, the coordinate transformation does not change the eigenvalues of extremely anisotropic metamaterials. In this regard, the extremely anisotropic metamaterial has the transformation-invariant feature [543, 544].

Based on the transformation-invariant nature, the chameleonlike devices can also preserve the adaptive property after transformation. Accordingly, a thermal chameleonlike rotator can be realized [544]. The transformed thermal conductivity is

(29){\kappa }^{\prime }= \left(\begin{array}{cc}{\kappa }_{rr}& {\kappa }_{rr}{\displaystyle \frac{r^{\prime }{\theta }_0}{R_2-R_1}}\\ {\kappa }_{rr}{\displaystyle \frac{r^{\prime }{\theta }_0}{R_2-R_1}}& {\kappa }_{rr}{\left({\displaystyle \frac{r^{\prime }{\theta }_0}{R_2-R_1}}\right)}^2+{\kappa }_{\theta \theta }\end{array}\right),

where {\theta }_0 is the rotation angle, and R_1 and R_2 are the outer and inner radii of the shell. The device transformed from an extremely anisotropic thermal conductivity can still rotate the heat flux without distorting the external temperature distribution, even though the background thermal conductivity changes [Fig.29(b)]. However, a normal rotator fails [Fig.29(c)]. In experiments, the radial thermal conductivity should be much larger than the background thermal conductivity, otherwise the chameleonlike rotator may fail. In other words, the chameleonlike rotator has a finite working range in practical applications. To improve the performance, the radial thermal conductivity needs to be enhanced.

10.3 Future outlook

Although the investigation of extremely anisotropic parameters has generated many essential results in a single physical field, the coupling of multi-physics fields introduces complexity, making further applications challenging. In particular, the simultaneous control of thermal and electromagnetic fields is crucial for applications, as on-chip circuits are significantly affected by external disturbances. Extremely anisotropic metamaterials have also great advantages in controlling multi-physics fields, which is a promising direction. Several studies have focused on the electromagnetic-thermal field [560–562] and the coupling between thermal conduction and fluid convection [563]. More multi-physics situations remain to be explored.

Metamaterials with a single function often lack flexibility in applications. Thus, designing metamaterials with multi-functions is essential to improve the performance. For instance, the thermal cloak-concentrator can automatically switch from cloak (or concentrator) to concentrator (or cloak) when the temperature changes [564]. Such a metamaterial can effectively manipulate the heat flux at will. The excellent properties of extremely anisotropic metamaterials make it a good choice to realize multi-functions in a single device.

Extremely anisotropic metamaterials are realized by combining several isotropic materials in practice. If the anisotropic thermal conductivity is particularly complex, analytical forward design might be infeasible, and the emerging technique of artificial intelligence becomes crucial. With the given ideal anisotropic thermal conductivity, artificial intelligence can help design a practical structure to implement that ideal parameter. We also do not need to worry about that the structure is too complex to fabricate because 3D printing can overcome those fabrication challenges. Therefore, combining artificial intelligence and 3D printing is promising to design various structures [565, 566].

10.4 Summary

We have concluded the properties of extremely anisotropic parameters through theoretical investigations from three perspectives. With transformation theory, extremely anisotropic parameters can be introduced naturally. The infinite thermal conductivity in a specific direction makes extremely anisotropic metamaterials an extraordinary choice for implementing directional heat transfer, which is the fundamental of various applications. With scattering cancellation theory, extremely anisotropic metamaterials are found with chameleonlike properties, indicating that they can spontaneously response to the change of the background thermal conductivity. The combination of these two theories further indicates the transformation-invariant nature of extremely anisotropic parameters, and hence, chameleonlike metamaterials with various functions can be designed, such as chameleonlike rotators. In the future, extremely anisotropic metamaterials can be developed to deal with multi-physics and multi-function issues. Their fabrication can be implemented by combining artificial intelligence and 3D printing. These results could provide fresh insight into advanced directional heat transfer.

11 Fractional models and thermodynamics of anomalous heat transport in low-dimensional systems

Shunan Li, Bingyan Cao^*

　 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

11.1 Background

Most macroscale heat transport in three-dimensional bulk materials obeys classical Fourier’s law of heat conduction, namely,

(30)q=-\kappa \mathrm{\nabla }T,

where q=q\left(x,t\right) denotes the heat flux, \kappa is the thermal conductivity, and T=T\left(x,t\right) is the locally defined temperature. In the absence of the internal heat source, Eq. (30) is usually combined with the following continuity equation:

(31){\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}}=c{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }t}}=- \left(\mathrm{\nabla }\cdot \mathit{q}\right),

where u=u\left(x,t\right) is the internal energy density, and c is the specific heat capacity per unit volume. For the constant material properties, combining Eqs. (30) and (31) will yield infinite propagation speeds of thermal perturbation [567], and to avoid such infinite propagation speeds, the non-Fourier constitutive models have been developed [568–571], for instance, the standard Cattaneo law [572] as follows:

(32)\mathit{q}+\tau {\displaystyle \frac{\mathrm{\partial }\mathit{q}}{\mathrm{\partial }t}}=- \kappa \mathrm{\nabla }T,

with \tau the relaxation time.

When it comes to the low-dimensional systems, heat transport is generally anomalous, whose signatures include the divergent length-dependence of the effective thermal conductivity, the divergent integral of the heat current autocorrelation function, the non-Brownian long-time behavior of the mean-square displacement (MSD), the nonlinearity of the steady-state temperature profile, and the fast relaxation of the spontaneous fluctuation [573–583]. The first three signatures often emerge from the fractional heat transport models, so we here concern these signatures. The divergent length-dependence of the effective thermal conductivity is defined in steady-state heat transport without the internal heat source, whose mathematical statement reads

(33)\underset{L\to +\mathrm{\infty }}{lim}{\kappa }_{eff}=+ \mathrm{\infty },{\kappa }_{eff}={\displaystyle \frac{|\mathit{q}|L}{|T(x=L)-T(x=0)|}}.

Here, {\kappa }_{eff} is the effective thermal conductivity, and L is the system length. In the one-dimensional systems, the length-dependence of the effective thermal conductivity commonly exhibits a power-law scaling {\kappa }_{eff}\propto L^{\alpha } with , and for the two-dimensional systems, the length-dependence becomes logarithmic, namely, {\kappa }_{eff}\propto \mathrm{l}\mathrm{o}\mathrm{g}L. The divergent integral of the heat current autocorrelation function can be stated as

(34)\underset{t\to +\mathrm{\infty }}{lim}{\displaystyle {\int }_0^t{C}_q\left(t^{\mathrm{\prime }}\right)\mathrm{d}{t}^{\mathrm{\prime }}=+\mathrm{\infty },}

where C_q=C_q\left(t\right) stands for the heat current autocorrelation function and typically scales as C_q\propto t^{\beta } with \beta \ge -1. This signature will lead to a time-dependent effective thermal conductivity in the Green-Kubo formula. More precisely, {\kappa }_{eff}\propto t^{\beta +1} for \beta >-1 and {\kappa }_{eff}\propto \mathrm{l}\mathrm{o}\mathrm{g}t for \beta = -1. Under the transit time approximation t\propto L, Eqs. (33) and (34) will be equivalent. The non-Brownian long-time behavior of the MSD refers to

(35)⟨\mathrm{\Delta }{x}^2(t\to +\mathrm{\infty })⟩\propto t^{\gamma },\gamma \in (0, 1)\cup (1,2 ],

where ⟨\mathrm{\Delta } x^2(t)⟩ is the MSD, and \beta is the so-called transport exponent. This signature can also be connected to other signatures. Utilizing the concept of the mean first passage time (MFPT), Li and Wang [576] have acquired \alpha = 2-2/\gamma. In Ref. [577], \beta =\gamma -2 is demonstrated based on Eq. (31), and this result does not assume any specific random walk model. Both \alpha = \gamma -1 and \beta = \gamma -2 indicate that the effective thermal conductivity is divergent in superdiffusive () and ballistic (\gamma =2) heat transport.

The heat transport models with the integer-order derivatives cannot give rise to the signatures mentioned above, which motivates the generalizations of Eqs. (30)−(32) produced by introducing the fractional-order derivative and integral operators [584–594]. The fractional heat transport models can cover the existing connections among the aforementioned signatures [595–597], and have been investigated based on the fractional Boltzmann transport theory [598–603]. Although the fractional heat transport models have attracted increasing attention, studies on thermodynamics of such models are not much. Existing results [602–604] imply that the fractional-order operators and the initial value terms will invalidate the framework of extended irreversible thermodynamics (EIT) for the heat transport models with the integer-order derivatives [605]. In the present work, we briefly summarize the relations between the fractional heat transport models and the aforementioned signatures, and discuss these operators the from thermodynamic perspectives.

11.2 Fractional models and anomalous signatures

The power-law divergence of the effective thermal conductivity, {\kappa }_{eff}\propto L^{\alpha }, can arise from the constitutive equations with the spatial fractional-order operators, i.e.,

(36)\begin{array}{rl}{q}_x=& -{\kappa }_{\alpha }\frac{{\mathrm{\partial }}^{1-\alpha }T}{\mathrm{\partial }{x}^{1-\alpha }}\\ =& -\frac{{\kappa }_{\alpha }}{2\mathrm{\Gamma }(\alpha )}{\int }_0^I\frac{1}{{\left|x-x^{\mathrm{\prime }\mathrm{\prime }}\right|}^{-\alpha }}\frac{\mathrm{\partial }T\left(x^{\mathrm{\prime }},t\right)}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\mathrm{d}{x}^{\mathrm{\prime }},\end{array}

(37)\begin{array}{rl}{J}_x=& -\frac{D_{\alpha }}{2\cos \left({\displaystyle \frac{\pi \alpha }{2}}\right)\mathrm{\Gamma }(\alpha )}\frac{{\mathrm{\partial }}^{1-\alpha }P}{\mathrm{\partial }{x}^{1-\alpha }}\\ =& -\frac{D_{\alpha }}{2\cos \left({\displaystyle \frac{\pi \alpha }{2}}\right)\mathrm{\Gamma }(\alpha )}{\int }_0^L\frac{1}{{\left|x-x^{\mathrm{\prime }}\right|}^{1-\alpha }}\frac{\mathrm{\partial }P\left(x^{\mathrm{\prime }},t\right)}{\mathrm{\partial }{x}^{\mathrm{\prime }}}\mathrm{d}{x}^{\mathrm{\prime }}.\end{array}

Here, q_x=q_x\left(x,t\right) is the one-dimensional heat flux, {\kappa }_{\alpha } is the generalized thermal conductivity, J_x=J_x\left(x,t\right) is the probability current, D_{\alpha } is the noise intensity, and P=P\left(x,t\right) is the probability density function (PDF) that is defined by

(38)P={\displaystyle \frac{⟨u(0,0)u(x,t)⟩-⟨u(x,t)⟩⟨u(0,0)⟩}{{\int }_0^I⟨u(x,0)u(0,0)⟩\mathrm{d}x-{\int }_0^I⟨u(x,0)⟩⟨u(0,0){⟩}^2\mathrm{d}x}}.

Eq. (36) is the nonlocal Fourier law [584], and Eq. (37) is equivalent to the Lévy−Fokker−Planck equation [597, 606–608]. Moreover, both of Eqs. (36) and (37) will produce nonlinear steady-state temperature profiles, whose boundary asymptotic behaviors fulfill

(39)\{\begin{array}{l}\underset{x\to 0^{+}}{lim}|T(x)-T(x=0)|\propto x^{\frac{\alpha }{2}}\\ \underset{x\to L}{lim}|T(x=L)-T(x)|\propto (L-x{)}^{\frac{\alpha }{2}}\end{array}

This means that the spatial fractional heat transport models also predict a connection between the power-law divergence of the effective thermal conductivity and the nonlinearity of the steady-state temperature profile.

As mentioned above, the heat current autocorrelation function scaling as C_q\propto t^{\beta } corresponds to {\kappa }_{eff}\propto t^{\beta +1} for \beta > -1. Such time-dependence of the effective thermal conductivity can be reproduced by the constitutive equations with the temporal fractional-order operators, for instance,

(40){\tau }^{\beta +1}{\displaystyle \frac{{\mathrm{\partial }}^{\beta +1}\mathit{q}}{\mathrm{\partial }{t}^{\beta +1}}}+\tau {\displaystyle \frac{\mathrm{\partial }\mathit{q}}{\mathrm{\partial }t}}=- \kappa \mathbf{\nabla }T,

where {\mathrm{\partial }}_t^{\beta +1} is the defined by the left Riemann−Liouville (RL) operator,

(41)\begin{array}{rl}\frac{{\mathrm{\partial }}^{\beta +1}\mathit{q}}{\mathrm{\partial }{t}^{\beta +1}}=& {}_0^{RL}{D}_t^{\beta +1}\mathit{q}\\ =& \frac{1}{\mathrm{\Gamma }(n-\beta -1)}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{t}^n}{\int }_0^t\frac{1}{{\left|t-t^{\mathrm{\prime }}\right|}^{\beta +2-n}}T\left(\mathit{x},t^{\mathrm{\prime }}\right)\mathrm{d}{t}^{\mathrm{\prime }},\end{array}

with n\in \mathbb{N}\bigcap (\gamma ,\gamma +1]. Eq. (40) is derived based on the Goychuk’s Boltzmann transport equation (BTE) [596, 609], and for phonon heat transport, the Goychuk’s BTE takes the form

(42){\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}}+v_g\cdot \mathrm{\nabla }f= {\tau }^{\beta }{\displaystyle \frac{{\mathrm{\partial }}^{\beta +1}}{\mathrm{\partial }{t}^{\beta +1}}}\left(f_0-f\right),

where f=f(\mathit{x},t,\mathit{k}) is the phonon distribution function, \mathit{k} is the wave vector, {\mathit{v}}_g is the group velocity, f_0= \frac{1}{\exp \left[\mathrm{h}\omega /(k_{B}T)\right]-1} is the Bose-Einstein distribution, h is the reduced Planck constant, and \omega is the angle frequency. Moreover, \gamma = \beta +2 can also be obtained from the Goychuk’s BTE [596], which agrees with Ref. [577].

The non-Brownian long-time behavior of the MSD can be predicted by the governing equations with the temporal fractional-order operators. The generalized Cattaneo equations (GCEs) [585] are paradigmatic examples. The GCE, GCE I, GCE II and GCE III are respectively given by

(43){\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }t}}+{\tau }^{\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }T}{\mathrm{\partial }{t}^{\gamma }}}=D{\mathrm{\nabla }}^{2}T ,

(44){\tau }^{\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }T}{\mathrm{\partial }{t}^{\gamma }}}+{\tau }^{2\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{2\gamma }T}{\mathrm{\partial }{t}^{2\gamma }}}=D {\mathrm{\nabla }}^{2}T ,

(45){\tau }^{\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }T}{\mathrm{\partial }{t}^{\gamma }}}+\tau {\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}}=D{\mathrm{\nabla }}^{2}T,

(46){\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }t}}+\tau {\displaystyle \frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{t}^2}}=D{\tau }^{1-\gamma }{\displaystyle \frac{{\mathrm{\partial }}^{1-\gamma }}{\mathrm{\partial }{t}^{1-\gamma }}}\left({\mathrm{\nabla }}^{2}T\right),

with D=\kappa /c the thermal diffusivity. The GCEs are proposed via introducing the temporal fractional-order operators into Eqs. (31) and (32), namely,

(47){\tau }^{\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }u}{\mathrm{\partial }{t}^{\gamma }}}=-\left(\mathrm{\nabla }\cdot \mathit{q}\right),

(48)\mathit{q}+{\tau }^{\gamma }{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }\mathit{q}}{\mathrm{\partial }{t}^{\gamma }}}=-\kappa \mathbf{\nabla }T,

(49)\mathit{q}+{\tau }^{\gamma }{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }\mathit{q}}{\mathrm{\partial }{t}^{\gamma }}}=-\kappa {\tau }^{1-\gamma }{\displaystyle \frac{{\mathrm{\partial }}^{1-\gamma }}{\mathrm{\partial }{t}^{1-\gamma }}}(\mathrm{\nabla }T).

In particular, Eq. (49) reproduces \gamma =\beta + 1 likewise [602].

11.3 Discussion from thermodynamic perspectives

Any fractional heat transport model must obey all of the laws of thermodynamics, and otherwise, this model is physically meaningless. Related issues have been addressed based on the fractional Boltzmann transport theory and Boltzmann−Gibbs (BG) statistical mechanics [602–604]. The results show that the continuity equation and the entropy production rate contain the fractional-order operators and the initial value terms in certain cases [602], i.e.,

(50){\tau }^{\gamma -1}{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }u}{\mathrm{\partial }{t}^{\gamma }}}=-\left(\mathrm{\nabla }\cdot \mathit{q}\right)+{\displaystyle \frac{1}{t^{\gamma }\mathrm{\Gamma }(1-\gamma )}}\left[{{\displaystyle \frac{{\mathrm{\partial }}^{\gamma }\left(\mathrm{\nabla }\cdot \mathit{q}\right)}{\mathrm{\partial }{t}^{\gamma }}}|}_{t-0}\right],

(51)\begin{array}{rl}\sigma =& k_B{\tau }^{\gamma -2}\int \ln \frac{f+1}{f}\frac{{\mathrm{\partial }}^{1-\gamma }\left(f_0-f\right)}{\mathrm{\partial }{t}^{1-\gamma }}\mathrm{d}\mathit{k}\\ & +\frac{k_B}{\mathrm{\Gamma }(\gamma -1)t^{2-\gamma }}\int \ln \frac{f+1}{f}\left[{\left(\frac{{\mathrm{\partial }}^{\gamma -1}f}{\mathrm{\partial }{t}^{\gamma -1}}\right)|}_{t-0}\right]\mathrm{d}\mathit{k},\end{array}

where \sigma =\sigma ( \mathit{x},t) is the entropy production rate. The continuity equation and the entropy production rate are associated with the first and second laws of thermodynamics, and with regards to the zeroth and third laws of thermodynamics, systematic discussion is still lacking. The zeroth law of thermodynamics states that if two systems are in thermal equilibrium with another system, these three systems are in thermal equilibrium with each other. Another statement of this law is that all systems in thermal equilibrium possess the same temperature. As a consequence, the zeroth law of thermodynamics will constrain the constitutive equations wherein the temperature is defined in the sense of equilibrium or local equilibrium. Mathematically, such constraint can be formulated as

(52)

which is valid for Eqs. (30) and (32). Unlike the heat transport models with the integer-order derivatives, the spatial fractional constitutive equations like Eq. (36), and the temporal fractional constitutive equations in some cases, i.e., Eq. (49) in the range of , will violate inequality (52). Thus, these fractional heat transport models are incompatible with the temperature defined in the sense of equilibrium or local equilibrium. We now focus on the third law of thermodynamics, which enforces the positive absolute temperature. However, even the standard Cattaneo law are not able to guarantee T>0 [610]. The standard Cattaneo law is actually a specific case of the temporal fractional heat transport models mentioned above, and hence, these models are not able to guarantee T>0 as well. In order to this remedy this deficiency, researchers [611–613] have suggested to state the third law of thermodynamics as

(53)\theta ={\left({\displaystyle \frac{\mathrm{\partial }s}{\mathrm{\partial }u}}\right)}^{-1}>0,

where \theta =\theta ( \mathit{x},t) is the so-called entropic temperature, and s=s(u, \mathit{q}) is the entropy density. Although the standard Cattaneo law can satisfy inequality (53) automatically, it is still unknown whether \theta > 0 holds for the fractional heat transport models.

11.4 Summary

Anomalous heat transport in the low-dimensional systems can be modeled in terms of the fractional heat transport models because the fractional-order operators can give rise to the signatures of anomalous heat transport, including the divergent length-dependence of the effective thermal conductivity, the divergent integral of the heat current autocorrelation function, the non-Brownian long-time behavior of the MSD, and the nonlinearity of the steady-state temperature profile. Furthermore, the fractional heat transport models also predict the connections between the aforementioned signatures. From the viewpoint of thermodynamics, the fractional heat transport models exhibit significant differences from the conventional heat transport models with the integer-order derivatives, which are mainly associated with the zeroth, first and second laws of thermodynamics.

12 Thermal null medium and its extensions

Yichao Liu, Tinglong Hou, Fei Sun^*

　 Key Lab of Advanced Transducers and Intelligent Control System, Ministry of Education and Shanxi Province, College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China

12.1 Background

Thermal metamaterials are engineered materials designed to control and manipulate heat flow in ways that surpass the capabilities of natural materials [614–617], which includes negative thermal conductivity [618, 619], extremely high effective thermal conductivity [620], self-adaptive heat transfer [621], and anisotropic thermal conductivity [622]. In recent years, thermal metamaterials have achieved successful applications in the field of heat flow control, including thermal invisibility [623–627], thermal illusions [628–631], thermal diodes [632, 633], remote heating [634], thermal concentration [635–637], and Janus functional thermal devices [638–640]

Among various thermal metamaterials, the thermal null medium (TNM) is a special highly anisotropic medium. For an ideal TNM, the thermal conductivity tends to infinity along the principal axis direction, while the thermal conductivity tends to zero in other directions perpendicular to the principal axis [641–644]. Due to the extremely anisotropic thermal conductivity of the ideal TNM, heat flow can only propagate along its principal axis, thus possessing perfect directional thermal conduction properties, which have significant application value in heat flow regulation. The following sections will summarize the development of TNM and highlight three key features of the TNM (see Fig.31): thermal surface transformation method based on directional heat guiding property, the transformation-invariant thermal conductivity under coordinate transformations, and the extensibility to simultaneously produce directional projection effects for heat fluxes and electromagnetic waves (i.e., multi-physical null media). This chapter will focus on introducing these three properties of the TNM.

12.2 Past to current development

12.2.1 The directional projection property of TNM and thermal surface transformation

The concept of null media initially emerged within the context of electromagnetic null medium [645–648] also known as optical void and nility medium [649, 650] and static magnetic null medium, which is also referred to as a magnetic hose [651–653]. These special media can be derived from extreme spatial stretching transformations by transformation optics [642, 647]. Subsequently, by employing similar extreme spatial stretching transformations and leveraging the principles of transformation thermodynamics [654–658], one can deduce the material parameters for a perfect TNM [642]: \kappa =\text{diag}(\mathrm{\infty },0,0), where “diag” indicates a diagonal matrix. This signifies that the thermal conductivity along the x-axis of the local coordinate system approaches infinity, corresponding to the local principal axis of the TNM, while the thermal conductivity in other directions approaches zero.

Due to the singularities in the parameters of an ideal TNM, its realization remains challenging. However, a unidirectional heat conduction effect can still be achieved with a reduced TNM that exhibits a thermal conductivity along the principal axis significantly higher than that of the background medium, and a thermal conductivity perpendicular to the principal axis significantly lower than that of the background medium. With the help of the effective medium theory [659–661], a reduced TNM can be equivalently realized by embedding a structure composed of staggered copper plates [with {\kappa }_{\text{copper}} =400\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})] and expanded polystyrene (EPS) boards [with {\kappa }_{\text{EPS}}=0.04\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})] into a background medium with moderate thermal conductivity, such as commercial thermal conductive pads with a thermal conductivity of 13 W/(m·K). This structure exhibits a thermal conductivity along the direction of the copper-EPS interface that is significantly greater than that of the background medium. In contrast, the thermal conductivity perpendicular to the interface between the two materials is significantly lower than that of the background medium, thus achieving directional heat conduction.

Based on the directional heat conduction characteristic of TNM, the theory of thermal surface transformation, i.e., a geometric design method, can be proposed [642]. The process of designing thermal devices using thermal surface transformation is as follows: First, design the shapes of the input and output surfaces of thermal devices. Next, establish a one-to-one correspondence between the points on these surfaces using geometric projection methods. Then, fill the space between them with TNM whose principal axis direction aligns with the geometric projection direction. This approach allows for the creation of a thermal regulation structure that corresponds to the projection of the temperature field distribution from the input surface onto the output surface. In preliminary research, this surface projection design method has been applied in designing thermal devices with various functions, including thermal cloaks [642], thermal concentrators [662], [663], thermal camouflage devices [664], thermal buffering shells [641], remote heating structures [665], thermal diodes [666], and heat focusing lenses [642].

12.2.2 The transformation-invariant property of TNM

Due to the thermal conductivity parameters of the TNM being extreme values—infinity and zero—if secondary or multiple coordinate transformations are performed within this type of medium, the principal values of zero and infinity remain unchanged after the transformation. That is, the material of TNM does not change through secondary or multiple transformations; only the principal axis direction of the TNM changes. Based on the material-transformation-invariant property of TNM under coordinate transformations, thermal devices can be designed in a concise and special manner. Many thermal devices that previously required coordinate transformation designs can now be directly designed by specifying the principal axis orientation of TNM, which includes, but is not limited to, thermal chameleon-like rotators [667], irregular-shaped thermal cloaks [668], thermal carpet cloaks [669], and directional heat transmission structures [670]. Therefore, by utilizing the material-transformation-invariant property of TNM, various heat flow control functions can be achieved using a highly anisotropic TNM, which can be practically realized through the interlacing of materials with high and low thermal conductivities. This property is not limited to TNM; it has also been applied to null media for other physical fields. For example, the material-transformation-invariant feature of optical null media have utilized this to design beam steering lenses [671], invisibility cloaks [672, 673], retro-reflectors [672], etc.

12.2.3 Thermal-electromagnetic null media and surface transformation that simultaneously control electromagnetic waves and heat flux

Staggered copper plates and EPS boards, which have been utilized as reduced TNM, can be extended to a double-physical-field null medium effective for both TM-polarized electromagnetic waves and heat flow by designing the spacing between the two materials to be sub-wavelength in size, and the length of the air channels enclosed by the copper plates to be an integer multiple of the operating wavelength. From a thermal conductivity perspective, for the layered structure with staggered copper and EPS sheets, due to the high thermal conductivity of copper and the poor thermal conductivity of EPS, it is equivalent to a reduced TNM with high thermal conductivity along the direction of the copper plate orientation and low thermal conductivity perpendicular to the copper plate orientation, which can guide the heat flow directionally along the orientation of the copper plates [674].

From the perspective of wave guidance, for microwave band electromagnetic waves, the complex permittivity of copper tends to infinity and can be modeled as a perfect electric conductor (PEC). Meanwhile, the permittivity of EPS is the same as that of air. Due to the shielding nature of copper plates as a PEC to electromagnetic waves, electromagnetic waves cannot penetrate the copper and the skin depth is nearly zero. As a result, electromagnetic waves are constrained to propagate along the EPS channels enclosed by the copper plates. The propagation direction of electromagnetic waves aligns with the orientation of the copper plates, enabling the structure to achieve the same directional guidance function for electromagnetic waves as a reduced electromagnetic null medium [674].

From the perspective of an effective medium, as long as the combined thickness of the copper plate and the EPS board is less than the wavelength of the incident electromagnetic wave, this structure can be represented as an equivalent medium. Its equivalent electromagnetic parameters can be seen as a simplified electromagnetic null medium effective for TM-polarized electromagnetic waves. That is, the permittivity tends towards infinity along the direction of the copper plate orientation, while the relative permittivity perpendicular to the copper plate direction is equal to the inverse of the EPS’s filling factor, and the relative permeability perpendicular to the copper plate direction is equal to the filling factor of the EPS [674]. These characteristics coincide with the electromagnetic material parameters of a reduced null medium effective for TM-polarized electromagnetic waves. To ensure the lowest reflectivity when the electromagnetic wave is incident on the structure and to avoid phase delay after passing through it, the length of the metal plate should satisfy the Fabry-Pérot resonance condition [675]. Due to the boundary conditions of electromagnetic theory, which require that the electric field can only have a vertical component at the PEC boundary, this structure can only act as a simplified electromagnetic null medium effective for TM polarization [675].

In summary, the structure obtained by the staggered arrangement of sub-wavelength intervals of copper plates and EPS boards can guide heat flow and electromagnetic waves simultaneously along the direction of the copper plate orientation. It can serve as a thermal-electromagnetic null medium (TENM) effective for both electromagnetic waves and heat flow, enabling simultaneous control and design of these dual-physics systems. [676, 677]. By utilizing the properties of TENM to simultaneously direct the guidance of electromagnetic waves and heat flow, one can achieve dual-physical field control structures with various functionalities. This is accomplished by designing the principal axis direction of the TENM. The method of designing electromagnetic-thermal structures through surface shaping and geometric projection is named as the thermal-electromagnetic surface transformation theory [676], which is a geometrical designing method and the extension of optical surface transformation [647, 648, 678]. In the thermal-electromagnetic surface transformation, the design process of electromagnetic-thermal devices can be described as a standard black-box design. The main steps can be summarized as follows: First, determine the shape of the input and output electromagnetic wavefronts and temperature field isotherms according to the pre-designed function of the device. Second, determine the shape of the input and output boundaries of the TENM, which can typically be designed to match the input/output wavefronts/isotherms. Third, find a proper one-to-one projection relationship that can geometrically project the points from the input boundary to the output boundary of the TENM one by one, ensuring that it also fulfills the function of the device and is easy to manufacture. Finally, fill the region where the projection tracks exist with the thermal-electromagnetic, ensuring that its principal axes coincide with the projection directions. By employing the thermal-electromagnetic surface transformation with the standardized design steps outlined above, numerous novel electromagnetic-thermal control structures have been proposed [676], including thermal-electromagnetic benders, thermal-electromagnetic multiplexers, thermal-electromagnetic cloak, and the thermal-electromagnetic splitter [679]. Based on the different physical laws that govern the propagation of physical fields, metamaterials can be divided into two major categories: diffusive metamaterials [614–617, 680] and wave metamaterials [681–686]. From the perspective of the differential equations, heat flows are determined from temperature distribution that satisfies Laplace’s equation, exhibiting diffusive propagation, while electromagnetic waves conform to Helmholtz’s equation, exhibiting wave propagation. Metamaterials effective for both heat flows and electromagnetic waves can simultaneously exert the same directional guidance and control effects on both wave and diffusion. This represents a step towards the integration of wave metamaterials and diffusive metamaterials, i.e., wave-diffusion-mixed metamaterials, which may have some potential applications in medical imaging and highly integrated chip systems [687].

12.3 Future outlook

For future research branches in the development of TNM, this chapter will also make predictions from three key features of the TNM: directional heat guiding property, transformation-invariant property under coordinate transformations, and the extensibility to multi-physical-fields controlling.

12.3.1 Future trends in the directional heat guiding property of TNM

The current utilizing of the directional projection properties of TNM is limited to the directional guidance of heat conduction, with no null media for heat convection and radiation guidance. If null media related to convection and radiation could be realized, it would be possible to introduce geometric design methods similar to surface transformations to achieve flexible control over convection and radiation. Just as thermal metamaterials are evolving from being effective for heat conduction to being effective for convection [688–691] and radiation [692–697], the development of TNM that is effective for convection and radiation, and even a generalized TNM effective for various heat transfer mechanisms, will be a direction for the future development of TNM. In addition, the current TNM is essentially a two-dimensional material, which can only guide heat directionally within the plane [666] or create an illusion of out-of-plane heat distribution [664]. From the perspective of implementation methods, the experimentally verified TNM primarily uses a structure of staggered copper plates and EPS boards, thus essentially guiding heat flow from line to line. In the future, developing three-dimensional TNM capable of point-to-point heat flow directional transmission will be another research branch for TNM.

12.3.2 Future trends in transformation-invariant property of TNM

Current research on the transformation-invariant feature of TNM primarily involves selecting a background space that contains the TNM region or a device composed of TNM for a specific function, such as cloaking or concentrating. Subsequently, secondary or multiple continuous coordinate transformations are carried out within the region occupied by TNM. This allows for changes in the position and shape of the device’s internal structure without affecting its external functionality and the materials it requires, i.e., TNM. Currently, all secondary transformations used in TNM are spatial coordinate transformations and do not include the time variable. Introducing time-related transformations into the secondary transformations within TNM may establish a correspondence between TNM and moving or time-modulated heat-conducting media. This could offer new solutions to time-related thermal modulating problems and spatio-temporal thermal metamaterials [698–705].

12.3.3 Future trends in the extensibility of TNM to multi-physical fields

The concept of null medium has been extended to media that are effective for both electromagnetic waves and heat flow, as well as for dual physical field control. Rapid development in this area is expected in the future. The nature of null media, which is easily expandable to multiple physical fields, will likely lead to two main research branches. One branch will focus on expanding the types and numbers of controlled physical fields, while the other will explore changes in control functions. Regarding the expansion of the number of physical fields, null media may evolve from the current thermal-electromagnetic [676, 677, 679]and acoustic-electromagnetic null media [674, 706–708], which are types of double-physical-field null media, to null media effective for three or more physical fields in the future. For instance, by utilizing a staggered arrangement of sub-wavelength intervals of copper plates, EPS boards, and air slabs in an appropriate combination, it may be possible to create a triple-physical-field null medium that is effective simultaneously for heat flux, acoustic waves, and electromagnetic waves.

Regarding the expansion of control functions, the current TENM provides the same directional guidance function for both electromagnetic waves and heat flow. Can we achieve distinct control for electromagnetic waves and heat flow? For instance, while directing electromagnetic waves, can we also efficiently dissipate heat? This could uncover more application value for TENM in the management of combined electromagnetic and thermal systems [709, 710]. For example, in electromagnetic functional structures containing metals, such as metallic metamaterials/metasurfaces, the presence of metal absorbs electromagnetic waves in the non-transmissive window of the structures, leading to electromagnetic-thermal issues. As the integration density of metamaterial units increases and the working time requirements for these structures grow, electromagnetic-thermal issues gradually become a limiting factor for the performance and stability of these structures. Electromagnetic-thermal issues are inevitable and can lead to problems such as short lifespans, poor stability, and limitations on incident electromagnetic power [711].

However, traditional heat dissipation structures cannot be directly applied to solve electromagnetic-thermal issues. Traditional heat dissipation structures are designed for a single physical field (i.e., the temperature field), and their design only needs to consider heat dissipation efficiency without considering the impact on the propagation of electromagnetic waves or the issues of thermo-electric coupling effects. For example, traditional heat dissipation structures are usually made of metals or other thermal conductors, which may interfere with the electromagnetic field and thus cannot be directly used to address electromagnetic-thermal issues. The combination of electromagnetic-thermal dual-physical-field null media [676] and active thermal metamaterials [618] can be extended to provide functions such as directional guidance of electromagnetic wave propagation while efficiently dissipating heat and other thermal control functions. This can offer new ideas for solving electromagnetic-thermal issues in metal-containing electromagnetic functional structures, serving as a multi-physical field compatible design scheme that takes into account both the electromagnetic performance and heat dissipation performance of electromagnetic functional structures.

12.4 Summary

The TNM, characterized by its unique anisotropic thermal conductivity, features directional heat guiding, transformation-invariant thermal conductivity under coordinate transformations, and extensibility to multi-physical fields, particularly in conjunction with electromagnetic waves. These characteristics have been successfully applied in designing various thermal devices, such as thermal cloaks, concentrators, and diodes, which exhibit controlled heat flux manipulation. The TNM has been extended to TENM, which merges thermal and electromagnetic wave guidance. This dual-physical-field control opens up new avenues for designing structures that can efficiently manage both heat conduction and electromagnetic radiation. Looking ahead, this chapter anticipates the expansion of TNM applications to include control over heat convection and radiation, moving beyond the current focus on conduction. It also foresees the development of three-dimensional TNMs for more advanced heat flow management. Moreover, integrating time-varying transformations could enable dynamic thermal modulation, offering innovative solutions for temporal thermal regulation. In summary, this chapter provides a comprehensive overview of the TNM, its evolution, and its extensions, outlining a roadmap for future research aimed at achieving advanced control in multi-physical field compatible thermal management.

13 Heat conduction in complex networks

Kezhao Xiong^1,*, Hang Dong¹, Zonghua Liu^2,*

　 ¹College of Sciences, Xi’an University of Science and Technology, Xi’an 710054, China

　 ²School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China

13.1 Background

In recent decades, much attention has been paid on the study of heat conduction in the case of one-dimensional (1D) atomic chain of the Fermi−Pasta−Ulam(FPU) model and it was shown that the heat conductivity diverges exponentially with the size of system, implying that the Fourier’s law does not necessarily hold true at the micro-nano scale [712]. However, in reality, heat transport is not only limited to 1D chain but often related to more complicated structures such as thermal interface materials and heat sink [713, 714]. These structures increase rapidly in recent years in the field of nanomaterials, called nanowire networks, where the phenomenon of “anomalous thermal transport” has also been observed. Very interesting, it is shown that the heat conductivity of systems composed of three-dimensional randomly arranged nanotubes is extremely low, although the heat conductivity of individual nanotubes can be very high [715], indicating that thermal transport in networked systems may have a significant difference from that in 1D chain.

It is well known that real systems, such as nanocomposites, biological systems, thin-film transistors, and artificial networks in nanosensors (nanotube/nanowire networks), are complex networks with topological structures fundamentally different from 1D and 2D lattices [716–718]. To study how heat is transported in complex networks, an initial effort was made in small size networks and it was found that a small size network can exhibit “strange” transport phenomena [719]. In particular, circulation of heat flux may appear in the steady state of a network of three oscillators only. To extend the study to a larger complex network, Liu and Li [720] made an investigation on simple networks of multiple chains with inter-chain couplings in 2007. They found that the two coupled particles between two chains will form an interface and introduce an interface thermal resistance, see Fig.32(a), which reduces heat current. Further, they found that this reduction of heat transport depends sensitively on the position and strength of the coupling, indicating that traditional methods for studying thermal transport in low-dimensional systems are not applicable to complex networks. That is, there are essential differences between electric networks transferred by electrons and heat conduction networks transferred by lattice vibrations, as the Kirchhoff second law cannot be applied directly to heat conduction networks without considering the interface resistances but can for the case of electric networks (circuits). In this sense, we need to systematically study heat conduction in complex networks, i.e., how the features of complex networks influence heat conduction. In the following sections, we will briefly summarize the development of this field and provide an outlook on future directions.

13.2 Past to current development

In a complex network, each node can be considered as an intersection of several chains and thus will induce interface thermal resistances. As different nodes of complex network have different node degrees, i.e., different numbers of connected chains, a diversity of interface thermal resistances among the chains exist, thus making it difficult to calculate the effective thermal resistance of the network. Therefore, a direct way to study heat conduction in complex networks is by numerical simulations. In 2010, Liu et al. [721] provided a first model to study heat conduction in complex networks by both theoretical analysis and numerical simulation, see Fig.32(b) for its schematic figure. They found that the topology of network significantly affects the transport and distribution of heat fluxes in the system. Their theory is based on the random matrix theory, which needs only the temperatures of nodes and the external flux of two thermal source nodes. By following this approach, Wang et al. [722] recently discussed the case of self-coupled loops, shown in Fig.33(a), and found that spatiotemporal thermostats can weaken the system’s total heat flux. This system of self-coupled loops can be used to control heat ratchet and flow reversal [723]. It was revealed that with the increasing of coupling strength, the total heat flow reverses and the heat flux in the self-coupled loop disappears. Additionally, there exists a critical coupling strength where vortex ratchet heat flux phenomena occur.

An important finding of heat conduction in complex networks is the effect of heat rectification [721], where the heat flux J_{ij} from node i to node j does not equal to the heat flux J_{ji} from node j to node i. To measure this effect, a rectification coefficient can be introduced as follows

(54)R_{ij}=\left|{\displaystyle \frac{J_{i\to j}-J_{j\to i}}{max\left(J_{i\to j},J_{j\to i}\right)}}\right|.

Larger R_{ij} corresponds to larger difference between J_{ij} and J_{ji}, i.e., larger rectification effect. To characterize the influence of network topology, we let R\left(\mathrm{\Delta }k\right) be the average of R_{ij} on those pairs of nodes with \left|k_i-k_j\right|=\mathrm{\Delta }k. Fig.33(b) shows the dependence of R\left(\mathrm{\Delta }k\right) on \mathrm{\Delta } k, where the curves with p = 1 and 0 represent the cases of random network and scale-free network [724], respectively, and the “up” and “down” panels represent the cases of without rewiring and rewiring the cluster coefficient to C=0.6 [725], respectively. The results show that network topology can seriously influence the effect of heat rectification.

At the same time, Volkov and Zhigilei [726, 727] considered the case of random nanotube networks. They assumed that the system’s thermal transport is solely controlled by the intertube contact thermal resistance and all the contact thermal resistances are equal, which can be considered as a simplified version of Ref. [721] with no interface thermal resistances. They provided an analytical equation for a randomly dispersed linear carbon nanotube (CNT) system, which was applied to a continuous network composed of CNTs. After that, many studies have been focused on the systems of CNTs, including the aspects of both theory and experiments [728–731].

These studies have discussed the influence of cross-links, cross-link mass, and varying sizes of carbon nanotubes on thermal transport in nanowire networks, but have not paid much attention to the aspect of network topology. Considering the fact that network topologies in real systems are diverse and important for their functions, it is essential to study how the network topology influences its functions. For this purpose, Zhu et al. [732] studied the structural characteristics of complex networks and found that the probability distribution function of the components of the representative eigenvectors can describe the localization on networks where the Euclidean distance is invalid. Recently, Shen et al. [733] discovered non-affine states in the elasticity of affine stacked-derived networks and revealed that the system exhibits a wide range of tunable Poisson’s ratios. Further, Jin et al. [734] designed disordered networks with a wide range of adjustable Poisson’s ratios based on triangular analysis. Although these two studies do not address thermal transport in networks, their models and methods provide insights to understand the vibrational characteristics between nodes of networks and thus can help us to understand thermal transport of networks.

Except these indirect studies, Xiong et al. [735] made a series of direct studies of thermal transport in complex networks. They presented a heat conduction model of complex networks to reveal the theoretical relationship between phonon spectrum width and node degree and explained how node degree influences interface thermal resistance [735], see Fig.33(c) for its schematic figure. In this model, each link represents a 1D FPU atomic chain and each node represents the intersection of different FPU atomic chains, in contrast to the model of Fig.32(b) with no atoms in each link. Then, they studied the effect of network asymmetry on thermal transport and found that asymmetric networks can show thermal rectification effects [736]. Further, based on a 2D regular lattice, Xiong et al. [737] presented another network model of thermal transport and found that a network can be transformed from a good conductor to a poor conductor by changing the network’s topology. They observed two interesting phenomena: (i) In some local links of the network, heat flows from lower temperature nodes to higher temperature nodes, named as “network thermal siphon effect”; and (ii) there exists an optimal network topology that exhibits a unique combination of low heat conductivity and high electrical conductivity, providing significant potential applications in thermoelectric materials.

As pointed out above, the interface thermal resistances are induced by the cross-links. To analyze their effects, a node mass model was presented [738, 739], where node mass is designed to depend on node degree. It was revealed that the heterogeneous distribution of nodes’ mass can make the thermal rectification be changed from negative to positive modes, as shown in Fig.33(d) where 1 and 2 represent the negative and positive rectification modes, respectively. Except the aspect in 1D atomic chains, other aspects of network topology may also influence heat conduction. For this purpose, the effect of degree correlation was discussed. It was found that the assortative networks enhance thermal transport in nanonetworks, while disassortative networks reduce thermal transport [740]. The aspect of controlling phonon band gaps has also been discussed and a pseudo-dispersion relation theory was established [741, 742], providing a theoretical foundation for understanding phonon vibrations and thermal transport characteristics in complex networks.

Moreover, it was found that there are heat flow localization and anomalous size effects in complex networks [743]. For the former, it was found that for various complex networks, their main heat fluxes are always localized within a subnetwork. While for the latter, it was revealed that in contrast to 1D atomic chains, the total heat flux of a complex network increases with the length of the atomic chains, indicating an anomalous size effect.

13.3 Future outlook

In recent years, the application of machine learning in materials science has been expanded, with notable progress in predicting material properties related to thermal transport. Phonons, as key carriers of thermal transport, have their density of states (DoS) as a crucial parameter for studying heat transfer. Although traditional theoretical calculations and experimental methods can determine phonon DoS, these methods often require extensive computational resources or experimental conditions. Chen et al. [744] proposed a machine learning model of the Euclidean neural network to predict the phonon DoS. This model can predict the DoS of 4346 unseen materials in less than 30 minutes by inputting only the atomic structure. The results showed that 70% of the test samples had a relative error of less than 10%. Although this method provides reference value for resource-limited experimental planning, relying solely on machine learning models for material property predictions is not realistic.

In addition to phonon DoS, machine learning also shows promise in the study of the thermoelectric properties of nanowire networks. Ebrahimibagha et al. [745] used multiple machine learning techniques to investigate the thermoelectric properties of polyaniline-carbon nanotube/nanowire networks. Through machine learning, the study analyzed how different structural and compositional variables affected the electrical conductivity, Seebeck coefficient, and thermal conductivity of the nanowire networks, providing important theoretical insights for designing efficient thermoelectric materials. To predict the thermal conductivity of polymer nanowire networks, Rabczuk et al. [746] proposed a hybrid machine learning method, combining artificial neural networks and particle swarm optimization. By selecting fiber thermal conductivity, Kapitza resistance, volume fraction, and aspect ratio as input parameters, the model significantly improved prediction accuracy, outperforming traditional neural networks. Similarly, Otara and Auta [747] used machine learning back propagation networks to predict the effective thermal conductivity of macroporous foam-fluid systems and found that porosity significantly influenced thermal conductivity. Moreover, Zhu et al. [748] employed convolutional neural networks from deep learning to establish a model for predicting the thermal conductivity of complex networks. The study analyzed the impact of network topology on thermal conductivity by connecting nodes through Delaunay triangulation and calculating thermal conductivities under different geometric topologies, which demonstrated the efficiency and accuracy of machine learning in predicting thermal conductivity in complex networks.

In summary, machine learning shows great potential in predicting the thermal transport properties of materials. Whether machine learning can be used to predict phonon DoS or study the thermal conductivity of complex networks is a challenging problem, but it is worthy to try.

Currently, we have achieved some understanding of the thermal transport properties and topological structures in complex networks, but extensive studies are still required. Specifically, there are five key topics:

1) There are many topological parameters to characterize a complex network, but so far only a few of them have been discussed in heat conduction, such as the node degree, clustering coefficient, and assortativity. Therefore, further research is needed to investigate the influence of the remaining topological parameters on thermal transport, such as the betweenness, community, and directional links.

2) It has been known that the substrate potential plays a crucial role in thermal transport in low-dimensional systems, but little attention has been paid to the case of complex networks.

3) It has been proved that the chaotic feature of low dimensional systems plays a critical role in heat transport, but it is unclear how this feature influences the thermal transport in complex networks.

4) In contrast to the pairwise interaction, the higher-order interaction is currently a hot topic in the field of synchronization, thus it is necessary to study how the higher-order interaction influences thermal transport in complex networks.

5) Currently, the rapid development of AI has made it possible to integrate machine learning with thermal transport in complex networks. Thus, AI is expected to be used in studying heat transport in complex networks.

13.4 Summary

So far, studies on thermal transport of complex networks have attracted more and more attention, which includes the aspects of models, numerical methods, theoretical analysis, optimization, and design, etc. With the rapid development of industries such as flexible materials, biomaterials, and nanomaterials, it is essential to explore how to effectively integrate the theoretical results on complex networks with the performance of heat conductivity in these materials. In this way, we may expect that in the near future, the features of thermal transport in complex networks can be used to guide the design of more effective functional materials and materials with unique thermal properties.

In summary, this chapter has briefly reviewed the progress of thermal transport in complex networks, i.e., from simple networks to complex networks. A main difference between the heat conduction in 1D chains and complex networks is that the former has no cross-links and thus no interface resistance, while the latter has cross-links and thus interface resistance. As these interface resistances depend on both the node degree and location in network, their influence to heat conduction is very complicated. Therefore, we need to study the thermal transport of complex networks from different aspects, including the node degree, clustering coefficient, assortativity, phonon spectra, phonon participation ratios, and pseudo-dispersion relations. Although significant progress has been made in studying thermal transport of complex networks, this field is still in its early stages. A potential application of these studies is in the aspects of nanomaterials, high-clustering materials, and artificial intelligence. Along this line, more effective functional materials and materials with unique thermal properties can be designed.

14 Thermal polaritonics: A novel and versatile heat channel

Sebastian Volz^*

　 Laboratory for Integrated Micro and Mechatronic Systems CNRS-IIS UMI 2820, The University of Tokyo, Tokyo 153-8505, Japan

14.1 Background

Thermal polaritonics is an emerging interdisciplinary field that combines classical thermal radiation and surface electromagnetic waves, particularly those associated with phonon−polaritons and plasmons. This field arises from the coupling of photons and phonons or photons and electrons, resulting in the formation of surface-bound electromagnetic waves known as surface polaritons. These waves can significantly influence thermal transport, especially at the micro and nanoscale, where traditional heat conduction mechanisms are often insufficient to describe the heat flux.

The concept of thermal polaritonics primarily focuses on the interaction of electromagnetic waves with the vibrational modes or the electrons of the materials, specifically those materials exhibiting phonon−polariton resonances and plasmonic behaviors. These interactions lead to the formation of surface phonon−polaritons (SPhPs) and surface plasmon−polaritons (SPPs), both of which are essential for understanding heat transfer in thin films and nanostructures.

One of the most significant contributions to the field was made by Professor Gang Chen, who, in a seminal paper published in Ref. [749], introduced the concept of surface phonon−polariton heat flux in silica nanofilms. His work demonstrated that the radiated heat flux in thin films (less than 100 nm in thickness) can dramatically increase at elevated temperatures (around 500 K), due to the enhanced role of surface phonon−polaritons. In bulk materials, the heat flux is limited by traditional radiative and conductive processes, but in thin films, surface-bound modes like SPhPs become dominant. This is because, in thin films, absorption of these surface modes decreases, and the surface polaritons can propagate over a broad range of wavelengths [750–752]. This contribution increases as the film thickness decreases and the temperature increases, offering exciting new possibilities for designing advanced thermal management systems at the small scales.

14.2 Past to current developments

14.2.1 Theory and experiments

The theoretical framework for understanding and quantifying phonon−polariton heat flux has evolved significantly over the years. Initially, much of the theoretical work focused on modeling the behavior of surface polaritons using basic electromagnetic theory. However, as the field progressed, more sophisticated methods were developed to account for the complex interactions in complex geometries [753].

Theoretical methods. Two primary theoretical approaches have been used to study the role of phonon−polaritons in thermal transport:

1) Maxwell’s equations and the Boltzmann transport equation: One of the most straightforward methods for studying phonon−polariton heat flux involves solving Maxwell’s equations to derive the dispersion relations for the surface phonon−polaritons. This approach also calculates the spectral propagation length and uses the photon Boltzmann Transport Equation (BTE) to estimate the thermal conductivity due to polaritons. This model is enhanced by coupling the BTE with the heat conduction equation to account for local absorption and emission of surface phonon−polaritons [750, 755, 759]. The photon-particle concept is here invoked to describe the two-dimensional polariton transport.

2) Electromagnetic Green functions and the fluctuation−dissipation theorem: Another approach uses electromagnetic Green functions, which provide a more detailed and accurate description of energy transfer between electromagnetic fields and thermal modes. By applying the fluctuation−dissipation theorem, one can derive the local Poynting vector, which quantifies the energy flux due to surface polaritons. While this method can be implemented for simple geometries [760], it often requires numerical methods to handle more complex systems. Advanced open-source software like JULIA or SCUFF-EM is increasingly being used for such simulations, enabling more accurate modeling of real-world systems.

Experimental methods. Experimental techniques to measure phonon-polariton heat flux have been developed alongside theoretical advancements. Traditional methods of measuring heat conduction, such as the 3\omega method and Time Domain Thermoreflectance (TDTR), can be implemented to detect the contribution of surface polaritons in thin films [756–758]. Further experiments using near-field probes and infrared cameras are used to detect the heat flux, as done in conventional thermal conductivity measurements, but including the contributions that arise from surface phonon−polaritons.

One challenge in experimental work is that the wavefront of the emitted polariton heat flux can spread significantly beyond the physical structure, which makes it difficult to measure directly with a local thermal sensor. To overcome this, experiments often include absorbers to capture the emitted heat flux and highlight the contribution from polariton modes. In addition to thin and waveguide films [761], works have also focused on polariton crystals [753, 754] micro-ribbons and wires [762-764], where the role of polaritons in heat transport has been shown to be significant.

14.3 Applications

Thermal polaritonics has a promising impact on several key applications in thermal management, particularly at the microscale and nanoscale. One of the most important applications is heat spreading in micro-opto-electronic systems, where managing heat at small scales is crucial for improving the performance and longevity of devices. The ability to control and manipulate polariton-based heat flux allows for the development of new heat transfer channels in these systems, which can greatly improve the efficiency of microelectronics.

Another promising application is in heat rectification, where the heat flux can be directed preferentially in one direction. The unique properties of surface polaritons, such as their wave-like nature and ability to propagate at nearly light speed, make them highly suitable for controlling heat flow. This capability could be used in a range of applications, from thermal management in microelectronics to novel devices in information processing and cryptography.

Polariton heat is also a very manipulable form of thermal energy. Since it behaves as a wave, with high propagation speeds and has the ability to be guided along surfaces and films [762], it opens up new possibilities for designing systems that can focus, steer, or control heat flux in ways that were previously not possible. This could lead to breakthroughs in a variety of fields, including thermal sensing, energy harvesting, and nanoscale cooling technologies [765].

14.4 Future outlook

The field of thermal polaritonics has seen rapid growth in recent years, with several key developments and breakthroughs. The waveguiding of polaritons in thin films and microstructures has been demonstrated, and there is growing evidence of their contributions to heat transport in metals [766], where plasmons play a similar role to phonon−polaritons. In addition, the study of near-field radiation between films has led to the hypothesis of Super-Planckian emission [767], and has uncovered new contributions of near-field [768] and edge [769, 770] modes in thermal radiation. These developments suggest that polariton-based heat transfer may enable more efficient thermal radiation beyond traditional limits.

In the future, there will be a continued focus on investigating two-dimensional (2D) plasmonic and phonon−polariton systems. These 2D materials, such as graphene, boron nitride and other van der Waals materials, offer exciting possibilities for controlling and manipulating surface polariton heat. The advent of new low-dimensional devices will likely lead to the discovery of new materials and systems where polariton heat can be harnessed in novel ways. In another direction, it is also remarkable that the contribution of phonon-polaritons to heat conduction in certain bulk materials was also recently uncovered [771].

14.5 Summary

Thermal polaritonics is an exciting and rapidly developing field [772] that combines fundamental principles of electromagnetism, phonon physics, and thermal transport. Theoretical advancements have provided a deeper understanding of how surface phonon−polaritons and surface plasmons contribute to heat flux in thin films and microstructures, while some mechanisms still remain to be enlightened [773]. Experimental techniques have been developed to measure these contributions, and applications in thermal management, heat rectification, and information processing are likely to take shape.

Looking forward, the field holds great promise, particularly with the exploration of 2D materials in polariton heat transport. As the theoretical and experimental tools continue to advance, we are likely to see new applications emerge in areas such as energy harvesting, nanoscale cooling, and advanced thermal management systems.

15 Radiative cooling with thermal metamaterials

Hengli Xie, Chunzhen Fan^*

　 Key Laboratory of Materials Physics of Ministry of Education, School of Physics, Zhengzhou University, Zhengzhou 450001, China

15.1 Background

With the rapid development of the global economy, energy demand continues to rise and accompanying energy consumption in cooling occupies a considerable proportion. Traditional cooling methods such as water cooling, wind cooling or air conditioners not only consume a lot of energy but also have adverse effects on the environment. This situation urgently requires us to explore new cooling technologies. Radiative cooling is a passive cooling method that relies on the emission of heat through the atmospheric window. As an innovative cooling method, radiative cooling does not require energy consumption and it is environmentally friendly [774]. This technology can efficiently manage heat and has been widely used in building cooling, electronic equipment heat dissipation, personal heat management, photovoltaic power generation, the aerospace field and medical field [775, 776].

According to Planck’s law, any object above absolute zero degree will emit radiation, particularly in the infrared spectrum [777, 778]. The Earth’s atmosphere possesses specific transparency windows from 8 μm to 13 μm, enabling thermal radiation to escape into outer space and thus cool the object [779]. As this radiative process continues, the surface temperature of the object gradually decreases [780–783]. Catalanotti et al. firstly investigated a spectrally selective emitter with high emissivity at night by coating a metal plate with a plastic layer in 1975 [784]. However, the demand for cooling during the daytime in practical application far exceeds that at night. Rephaeli et al. proposed a metal-dielectric structure with a silver layer as the reflector, which generated a net cooling power of 100 \mathrm{W}/{\mathrm{m}}^2 under direct sunlight theoretically [785]. Thereby it greatly advanced the application of radiative cooling. In 2014, Raman et al. experimentally designed a seven-layer photonic structure on a silver layer, achieving a cooling effect of 4.9 {}^{\circ }\mathrm{C} [786]. Since then, a series of studies have been conducted to achieve efficient cooling. Up to now, cooling designs typically fall into four categories in Fig.35, namely, the metamaterials, random particles, multilayers, and porous polymers.

Among them, thermal metamaterials typically comprise subwavelength units, enabling the manipulation of thermal radiation beyond the constraints of natural materials to achieve efficient cooling [787]. The constituent materials with intrinsic high absorption in the atmospheric window is preferred, such as silicon nitride, amorphous silicon, SiO₂ microspheres, Al₂O₃, HfO₂, PVDF-HFP, and PDMS. Meanwhile, metallic layer is frequently employed to reduce the solar influence with novel metals (Au, Ag, Al, Cu). To further enhance emissivity in the atmospheric region, materials are often arranged in multilayers with finely tuned thickness or sequence, employing metal−insulator−metal designs, combining nanoparticles in polymers or porous structures, creating patterned surfaces [786, 788, 789]. Recently, by incorporating materials like graphene, liquid crystals, and phase change materials, active thermal metamaterials introduce additional degrees of freedom, enabling dynamic control over thermal emission. For instance, the exceptional electrical and thermal conductivity of graphene can be leveraged to modulate its response to external stimuli, while liquid crystals and phase change materials offer reversible transformations between different states, providing adaptive functionalities.

Thermal metamaterials are engineered at nanoscale to possess properties not found in natural materials, thereby enhancing their heat management capabilities. Currently, the applications of radiative cooling are being continuously developed and expanded to include building cooling, solar cell cooling [790, 791], personal thermal management [792–794], colored radiative cooling [795, 796], water condensation or collection [797, 798], thermoelectric generators [799–802], and space cooling [803, 804]. Meanwhile, machine learning in the design of thermal metamaterials and topological thermal metamaterials in radiative cooling are also developed. In this chapter, we offer a comprehensive study of radiative cooling principles aimed at enhancing reflectance in the solar band and emissivity in the atmospheric transparency band. We also analyze various thermal metamaterials for radiative cooling designs and applications, identify their shortcomings and propose potential solutions. The following sections will provide a brief summary of the development of this field from both theoretical and application perspectives, offering an outlook on future directions.

15.2 Past to current development

15.2.1 Principle

Thermal metamaterials are tailored to exhibit high emissivity within atmospheric transparency windows, while minimizing emissivity at other wavelengths to prevent the re-absorption of radiated heat. The cooling effect is commonly assessed through metrics such as cooling power or the reduced temperature difference. The heat exchange process between the radiative cooler and cold space adheres to the energy balance in Fig.36. The governing equation of the radiative cooler follows

(55)c_c{m}_c{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}t}}=S\times P_{net},

where c_c is the heat capacity of the radiative cooler, m_c is the mass of the radiative cooler. T is the surface temperature of the radiative cooler. t is the time. S is the area of the radiative cooler. The net cooling power (P_{net}) of the radiative cooler incorporates four parts, namely, the surface emission power (P_{rad}), the absorbed solar energy (P_{sun}), the atmospheric thermal radiation (P_{atm}) and the parasitic heat loss (P_{loss}). They observe the following equations,

(56)\begin{array}{rl}{P}_{net}(T)=& P_{rad}(T)-P_{atm}(T_{amb})-P_{sun}\\ & -P_{loss}(T,T_{amb}),\end{array}

where T_{amb} is the ambient temperature. The radiated power from the structure can be expressed as

(57)P_{rad}(T)= S{\displaystyle \int \mathrm{d}\mathrm{\Omega }\cos \theta {\displaystyle {\int }_0^{\mathrm{\infty }}\mathrm{d}\lambda I_{BB}(T,\lambda )\epsilon (\lambda ,\theta ),}}

where \int\mathrm{d}\mathrm{\Omega } = 2\pi {\int }_0^{\pi /2}\mathrm{d}\theta\sin \theta is the angular integral over a hemisphere. I_{BB}(T, \lambda) is the radiation intensity of a blackbody at a temperature of T. According to Planck’s law, it can be written as

(58)I_{BB}(T,\lambda )={\displaystyle \frac{2h{c}^2}{{\lambda }^5}}{\displaystyle \frac{1}{{\mathrm{e}}^{\frac{hc}{\lambda k_{B}T}}-1}},

where h is Planck’s constant, k_B is Boltzmann’s constant, c is the speed of light and \lambda is the wavelength. \epsilon (\lambda ,\theta ) is the emissivity of radiative cooler surface in the atmospheric transparency window. To maximize P_{net}, the ideal emissivity of the atmospheric window of 100% is required. P_{atm} is the absorption power of atmospheric thermal radiation. It is written as

(59)P_{atm}(T_{amb})=S{\displaystyle \int \mathrm{d}\mathrm{\Omega }\cos \theta {\displaystyle {\int }_0^{\mathrm{\infty }}\mathrm{d}\lambda I_{BB}(T_{amb},\lambda )\epsilon (\lambda ,\theta ){\epsilon }_{atm}(\lambda ,\theta ),}}

The emissivity of the atmosphere is {\epsilon }_{atm}(\lambda ,\theta ) = 1-t(\lambda {)}^{1/\cos \theta }, where t(\lambda ) is the atmospheric transmittance in the vertical direction [811]. In addition, the atmospheric emissivity is closely related to the water vapor content, carbon dioxide, ozone layer, and other molecules [812].

The absorbed solar energy can be expressed as

(60)P_{sun}=S{\displaystyle {\int }_0^{\mathrm{\infty }}\mathrm{d}\lambda \epsilon (\lambda ,{\theta }_{sun})I_{AM1.5}(\lambda ),}

where I_{AM1.5}(\lambda ) is the solar illumination of the AM1.5 spectrum. It is closely related to the location and atmospheric conditions [813].

The power loss caused by convection and conduction is represented as

(61)P_{loss}(T,T_{amb})=h_c(T_{amb}-T),

where h_c is the non-radiative heat transfer coefficient, which represents the heat loss between the cooler and the environment due to convection and conduction. h_c can be evaluated by testing wind speed (v) in the form of h_c = 1.44v+4.955. In the outdoor environment, it is usually taken from 0 to 5 m/s with a moderate wind speed [814].

Thermal emission of the radiative cooler strongly depends on the average reflectivity ({\overline{R}}_{sun}) in the solar band (0.25−2.5 μm) and the average emissivity \overline{\epsilon }({\lambda }_1,{\lambda }_2) in the first and second atmospheric windows. They are defined as

(62){\overline{R}}_{sun}={\displaystyle \frac{\underset{0.25\text{μm}}{\overset{2.5\text{μm}}{\int }}{I}_{sun}(\lambda )R(\lambda )\mathrm{d}\lambda }{\underset{0.25\text{μm}}{\overset{2.5\text{μm}}{\int }}{I}_{sun}(\lambda )\mathrm{d}\lambda }},

(63)\overline{\epsilon }= {\displaystyle \frac{\underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}\epsilon (\lambda ,\theta )I_{BB}(T,\lambda )\mathrm{d}\lambda }{\underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}{I}_{BB}(T,\lambda )\mathrm{d}\lambda }}.

15.2.2 Application

The enhanced emissivity of thermal metamaterials in the infrared region can be meticulously controlled through top-patterned subwavelength unit cells. Currently, the primary designs of thermal metamaterials in radiative cooling encompass multi-layer films, nanopatterned arrays, metal/dielectric composite structures, porous materials, phase change composites and topological insulator structures. The integration of these advanced materials facilitates precise tailoring of the emission spectrum. The objective is to maximize emissivity within the atmospheric window while minimizing it elsewhere to prevent the re-radiation of heat back into the environment. This optimization results in highly efficient radiative cooling, surpassing the capabilities of conventional materials. Subsequently, we will delve into the application of thermal metamaterials in various domains, including building cooling, colored radiative cooling, personal thermal management, photovoltaic (PV) cooling and power generation, water harvesting, space cooling and topological thermal metamaterials, among others.

Cooling the building. Radiative cooling offers a passive and sustainable method to maintain comfortable indoor temperatures in buildings. To enhance the emissivity of radiative cooling in the transparency window, it is necessary to analyze deeply both the constituent material and the structural design. Selecting a material with intrinsic extinction coefficient in the atmospheric window band is critical. Nanoparticles such as SiO₂, ZrO₂, Al₂O₃, TiO₂, BaSO₄, CaSO₄, Y₂O₃, and h-BN, have been demonstrated to enhance the reflection based on the Mie scattering theory [815]. Furthermore, cooling performance of the porous structure is also influenced by their size, shape and volume fraction. Polymers for radiative cooling include PVDF-HFP, PMMA, PDMS, POE, PTFE and others [816–820]. With the development of nanofabrication technology, the construction of metamaterials with subwavelength unit cells greatly improve the thermal emission [821, 822]. The square [803, 823], cross [824], hollowzigzag [825], pyramid [789], triangular [826], hyperbolic geometries [827] have been carefully designed to enable significant resonances. A north-facing low-angle radiative cooling-film roof yields the best radiative cooling performance by using spectrally selective metamaterial-based building envelopes [828]. Improved model was developed to evaluate a metamaterial-based new cool roof performance, which can significantly reduce roof-induced cooling load [829]. Actually, the location is another factor to determine the cooling behavior of the design [797].

Personal thermal management. The development of personal thermal management systems leveraging thermal metamaterials is an innovative field that focuses on enhancing individual comfort and energy efficiency [830]. Radiative cooling textiles are engineered to manipulate heat transfer in unconventional ways, enabling efficient cooling without the need for conventional energy-intensive methods. Hybrid metamaterial textiles with a polymer-based nanophotonic textile for passive personal cooling indoors and outdoors were systematically carried out [831]. A multilayer metafabric knitted with composite microfibers that incorporates hierarchically designed random metamaterial structures to directly integrate radiative cooling technology for personal thermal management [793]. To endow it more flexibly control on the personal thermal management, smart clothing with adaptive functionality of a hierarchically programmed meta-louver fabric with switchable modes was developed [832]. Efforts are ongoing to integrate these metamaterials into wearable technologies by designing flexible, breathable, and aesthetically pleasing fabrics that incorporate the cooling properties [788].

Colored radiative cooling. Based on the assumption of maximum cooling power, it is necessary to conceive colored radiative coolers to alter their functional and esthetic characteristics. Thus, integrating color display and radiative cooling is necessary to meet both aesthetic and practical requirements in real-life applications. A colored radiative cooler with a selective emitter is stacked on top of two nanocavities. By exploiting the coupling effect between two optical resonances inside the stacked nanocavities, the color display can be modified without impairing the emission in the atmospheric window [833]. Quartz-SiC thermally emitting photonic crystal layers, conical metamaterial, double-layer patterned metamaterials, Si nanowires as the Si nanostructure and other has been developed to balance the color appearance and the radiative cooling effect [834]. We have proposed an efficient colored radiative cooler with cross-shaped SiO₂ unit cells [795] and by depositing a cone emitter on an elliptical metal-insulator-metal reflector, the colored radiative cooler can be tuned by the thickness of the dielectric layer and polarization [796]. The addition of water-based colorants into the coating allows for a wide range of color options without compromising its properties [835].

Photovoltaic cell cooling and power generation. The integration of radiative cooling with both photovoltaic (PV) systems and thermoelectric generators (TEGs) have drawn much attentions. This combination aims at maximizing energy efficiency by utilizing waste heat from PV panels and converting it into usable electricity via TEGs, while the radiative cooling system helps maintain the PV cells at an optimal temperature for enhanced performance. For example, radiative cooling film was used to replace the heat sink as the cold side of thermoelectric generator TEG to create a lower cold-side temperature than the atmospheric [836]. It was experimentally verified by power generation from radiative cooling of a PV cell at night using a TEG module [837]. We have designed a thermal metamaterial with a grating-textured top layer on the bottom multilayer constitutes of MgF₂/SiO₂/Al₂O₃/TiO₂ sequentially. It could lower the working temperature and enhance the efficiency through radiative cooling [807]. To endow it with active tunable ability, switchable between photothermal and radiative cooling based on self-adaptive VO₂/PDMS metamaterial can also be achieved with the phase transition of the vanadium oxide [838]. Recently, Sub-ambient daytime radiative cooling and PV power has been carried out with a transparent radiative cooler atop a PV cell. It experimentally demonstrated passive cooling 5.1 {}^{\circ }\mathrm{C} below air temperature under 1000 W/m² solar intensity, alongside PV power output of up to 159.9 W/m² from the same area [839].

Radiative cooling with water condensation or harvesting. Radiative cooling presents a promising avenue to address the water scarcity issues. By reaching temperatures below that of the surrounding atmosphere, it facilitates the condensation of water vapor, effectively functioning as an innovative, energy-free method of condensation water harvesting. Thermal metamaterial with polar dielectric microspheres randomly in a polymeric matrix was fabricated, it had an infrared emissivity greater than 0.93 across the atmospheric window [788]. A selective radiative cooling film was applied on top of a shallow aluminum box that contained with an enclosed body of water for daytime radiative cooling [797]. To improve the performance of solar panels with thermal metamaterials, thermal emissivity can be well engineered, thereby both the amount of water production and the suitable temperature and humidity range can be significantly improved. It can be beneficial for both nighttime water harvesting and daytime solar cell efficiency enhancement [798]. In addition, the selection of the ideal thermal metamaterials emitters for the efficient water harvesting has been investigated in our study [840].

Space radiative cooling. Radiative cooling technology plays a vital role in space applications, primarily due to its ability to effectively dissipate heat without consuming any energy. Satellites in space are subjected to the direct exposure of solar radiation and extreme temperature variations of the cosmic background. Radiative cooling materials assist in regulating the satellite’s surface temperature, preventing electronic components from overheating, and thus ensuring the long-term stable operation of the satellite. An aluminum-doped zinc pxide metasurface on a SiO₂ dielectric layer on an Al mirror was designed for radiative cooling of spacecraft. It achieved an emissivity of 0.79 in the thermal infrared through the broad plasmonic resonances with an enhanced absorption of electromagnetic radiation [803]. A multi-material thin film was optimized through a genetic algorithm was developed to encounter the harsh space environment of the external surface of spacecraft [804]. Recently, the active smart thermal emitter of VO₂ metasurface with high visual transparency for passive radiative cooling regulation in space and terrestrial applications were elaborately investigated [841]. A design with Janus optical properties in the mid-infrared region, consisting of the nanoporous polyethylene films, the polyethyene oxide film, and silver nanowires was explored for low-temperature space cooling [842].

In addition, the design of thermal metamaterials using machine learning to achieve radiative cooling effects is a cutting-edge and promising approach. This method combines materials science, thermodynamics, and artificial intelligence techniques aimed at creating new materials that can efficiently dissipate heat. Utilizing machine learning for the design of thermal metamaterials has the potential to radically transform how we design and manufacture highly efficient heat dissipation materials. A daytime radiative cooling emitter consisted of polydimethylsiloxane, silicon dioxide, and aluminum nitride from top to bottom on a silver-silicon substrate was designed by using machine learning and genetic algorithms for daytime radiative cooling [843]. A data-driven approach with biomimetic photonic structural materials speeds up the design and optimization of radiative cooling metamaterials [844]. A transmissive colored radiative cooling film was first designed by using a mixed-integer memetic algorithm on top of a nanocavity-based color filter [845]. Researchers have successfully designed and synthesized a series of thermal metamaterials with unique topological structures [846]. They hold promise for playing a significant role in high-efficiency heat dissipation components, low-energy consumption cooling systems, intelligent temperature-control clothing, and other areas, bringing revolutionary changes to multiple industries [847]. Inspired by the nature animals, biological thermo regulation schemes have contributed to our interest in radiation-cooled bionic coatings [848]. Multi-bioinspired flexible thermal emitters for all-day radiative cooling and wearable self-powered thermoelectric generation resulted in a remarkable cooling performance of 7.3 °C during the day and 10.2 °C at night [849].

15.3 Future outlook and summary

Radiative cooling with thermal metamaterials revolves around manipulating thermal radiation through carefully designed materials and structures, leveraging phenomena like surface plasmon resonance and incorporating active materials for enhanced thermal emissions. This innovative approach opens up new avenues for efficient and sustainable cooling solutions across various industries. Ongoing research aims to further refine these materials and explore new active components to find what’s possible in thermal management.

Although the radiative cooling technology has accomplished some progress [850, 851], there are still some challenges in terms of performance improvement and practical application. The commercialization of radiative coolers requires not only consideration of the cost, optical, and mechanical properties of the material, but also the development of large-scale and simple manufacturing processes, such as brushing and spraying [852]. Moreover, the thermal stability, corrosion resistance, and hydrophobicity of radiative coolers are critical in complex and changing outdoor environments where their lifetime is at stake [853]. These issues need to be resolved in future research. Radiative cooling technology currently faces several major challenges, including its relatively low cooling power around 100 W/m² and a high dependence on atmospheric transparency. This means that the effectiveness of radiative cooling can be significantly diminished under cloudy or rainy weather conditions. Furthermore, various geographical locations, seasonal changes and environmental factors also have a substantial impact on the cooling power. As a result, the search for universally applicable radiative cooling materials remains an important research direction. Most of the current radiative coolers exhibit broadband emission with uniformly high emissivity throughout the infrared band, while selective emitters exhibit high emissivity only in specific atmospheric transparency windows and maintain high reflectivity in other bands. This trade-off on the emissivity between broadband and selective emission efficiency presents a challenge in optimizing radiative cooling systems for practical applications [854]. In addition, although the metal layer can effectively reflect sunlight, its high cost may increase the production cost of the product, thereby limiting its large-scale application. The random particle structure exhibits poor environmental stability and some particles may agglomerate or degrade under different environmental conditions, thereby affecting its scattering performance and the stability of the reflectivity. Although the porous structure can achieve high reflectivity through multiple scatterings, its preparation process is relatively complex and requires precise control of synthesis, which may increase the difficulty and cost of production.

In the realm of personal thermal management, developing fabrics that can effectively dissipate heat during high temperatures while providing excellent insulation during cold temperatures continues to present significant challenges. Integrating radiative cooling films with thermoelectric generator systems efficiently to ensure stable operation under different environmental conditions, while optimizing energy conversion and heat dissipation mechanisms to achieve maximum output power from solar cells is crucial. The appearance of vivid color inevitably causes a decrease in the reflectivity of radiative coolers in the solar band and a reduction in cooling power [796, 809]. Therefore, it is crucial to establish a balance between color display and cooling power to promote the application of radiative coolers. Although color has been obtained by utilizing dyes molecules, chalcogenide light-emitting particles or constructing special photonic structures, the application scenarios are fairly limited [795, 855]. Smart materials are available to fulfill different aesthetic needs while minimizing the impact on cooling performance in the application of color radiative cooling devices. In dynamic environments, active radiative coolers should have the ability to switch cooling on and off automatically as the ambient temperature varies. At high temperatures, the cooler requires a large solar reflectance and high infrared emissivity to achieve cooling, while at low temperatures, the infrared emissivity should be automatically reduced to prevent heat loss. Hence, the development of adaptive radiative coolers that can automatically adjust their performance according to environmental temperature changes and provide appropriate temperatures under different temperature conditions will meet the needs of human beings. At present, it has become a mainstream research direction to regulate solar reflection and infrared emission by utilizing phase change materials [856]. However, common phase change materials of VO₂, Ge₂Sb₂Te₅, Sb₂S₃ face many challenges in practical applications and are difficult to commercialize due to their high phase change temperatures [857]. In contrast, thermochromic hydrogels may be a better choice, which can undergo phase transition in a lower temperature range, thus better adapting to different ambient temperatures [858, 859]. Additionally, combining radiative cooling films with condensation systems to enable efficient seawater desalination and collection is another area of importance. Especially in the extremely complex conditions of outer space, comprehensively considering various environmental factors and developing materials and technologies suited for these conditions remains a critical issue that urgently needs resolution. With ongoing development, the machine learning and topological thermal metamaterials are further developed, the time consuming is needs to be improved and the experimental realization is still needing to be carried out, instead of the present the theoretical study.

In summary, we have introduced the development of radiative cooling with thermal metamaterial and the basic principle is also provided. To engineer the thermal emission in the atmospheric transparency window, the minimum influence of the solar effect should be considered. The engineered emissivity can be well optimized with the careful design of the subwavelength unit cell to manipulate the surface resonance. For the application of radiative cooler with thermal metamaterials, the high emissivity is highly required with simple, cost effective, durable, stability. Looking forward to the future development direction, radiative coolers are expected to play an important role in fields such as building energy conservation, personal thermal management, seawater desalination, promoting the realization of sustainable development goals. With the continuous progress of fabrication technology and the development of materials science, the wide application of radiative coolers will provide an effective solution to address the global energy crisis and climate change. It also meets the goals of peak carbon dioxide emission and carbon neutrality.

16 Progress of twist angle-controlled radiative near-field thermal transfer

Xinran Li, Yungui Ma^*

　 State Key Laboratory of Extreme Photonics and Instrumentation, Centre for Optical and Electromagnetic Research, College of Optical Science and Engineering; International Research Center (Haining) for Advanced Photonics, Zhejiang University, Hangzhou 310058, China

16.1 Background

Facilitated by the tunneling of evanescent waves and the enhanced localized optical density states [860], near-field radiative heat transfer (NFRHT) between two closely spaced surfaces can exceed far-field blackbody radiation by several orders of magnitude, especially between two surfaces with strong light-matter interaction in the thermal infrared bands. Fundamental studies were initially focused on the near-field transfer between bulks, indicating the important role of bounded modes such as SPhPs [861], surface plasmon polaritons (SPPs) [862], or frustrated modes [863]. However, advancements in emerging materials such as polar solids [864], two-dimensional (2D) materials [865] and even metamaterials [866, 867] have elevated the study of near-field thermal transfer to a new level. Additionally, numerous potential applications including thermal management [868], energy harvesting [869], radiative cooling [870], and near-field imaging [871] have been identified.

In recent years, the field of twistronics, following the pioneering work on twisted bilayer or multilayer photonic templates, especially the bilayer graphene’s moiré patterns [872], has inspired significant advancements in photonic structures. This twist-induced modulation method have attracted considerable attention and proven to be a powerful platform for the development of innovative photonic devices [873]. More recently, this concept has been applied to the studies of NFRHT to explore the tunability of photon tunneling through modulated mode coupling [874–878, 880, 888]. This approach holds potential applications such as 2D thermal transistors [889] and nonreciprocal thermal switchers [886]. The relative twist between two heat transfer surfaces has been demonstrated as an effective solution for modulating NFRHT. The subsequent sections will provide a concise overview of the twist-induced NFRHT development from natural materials, metamaterials, and cases controlled by external energy, and will offer insights into the future research directions.

16.2 Past to current development

16.2.1 Natural materials

The near-field twist-angle controlled heat transfer between natural materials is usually regulated by the in-plane anisotropy. On the one hand, the anisotropic modes between the two surfaces can be mismatched in the momentum space with the increase of relative twist angle, particularly, when the twist angle equals to 90°, the mismatch reaches the maximum, significantly suppressing the heat transfer compared to the aligned case. Materials based on this manipulation mechanism include 2D materials such as uniaxial Vander Waals materials (hBN) and Weyl semimetals (Co₃Sn₂S₂) with anomalous Hall effect. Three-dimensional (3D) materials like uniaxial (calcite, \alpha-sapphire and \alpha-quartz) and biaxial crystals (\alpha-MoO₃) also result in a high modulation performance. On the other hand, the relative rotation between the nanostructures in the near field leads to extremely attractive phenomenon. Unlike in electric systems, there is no “magic angle” for sudden changes in heat flux in current natural bulky heat transfer systems, which will be discussed below.

Vander Waals materials such as hBN [874] are commonly used natural anisotropic materials. The dielectric function perpendicular and parallel to the surface satisfies real({\epsilon }_{\mathrm{\perp }}) \times real({\epsilon }_{||}) < 0 results in the excitation of the hyperbolic modes. These modes support large inplane wavevector, enhancing the tunneling of evanescent waves by several orders of magnitude. However, with the relative rotation, coupling mismatches between the two surfaces begin to appear, restraining the heat transfer in this case. In addition, uniaxial crystals such as \alpha-quartz [875], \alpha-sapphire and calcite’s special dielectric dispersion show a similar characteristic. The hyperbolic modes and even anisotropic double-negative permittivity modes play a crucial part, which can be easily applied to the spatial modulation.

However, modulation by adjusting the relative twist angle between the thermal surfaces can be possible only when the optic axis is in-plane oriented, which poses a challenge in the growth of the materials. Biaxial crystals such as \alpha-MoO₃ [876] make both in-planes and out-of-plane optic axis can easily overcome this problem, providing the potential to effectively modulate the NFRHF. The heat flux dependent on the twist angle varies significantly along different crystalline directions, and the modulation contrast can reach a factor of two when the heat flux is aligned along the [010] direction. This substantial modulation contrast mainly attributed to the misalignment of anisotropic hyperbolic surface phonon polaritons and hyperbolic phonon polaritons between the two surfaces.

Compared to non-magnetic material, the anisotropy mechanism in magnetic material are different. For example, in Weyl semimetals [877] either inversion or time-reversal symmetry needs to be broken to split a doubly degenerate Dirac point into a pair of Weyl nodes with opposite chirality. This split alters electromagnetic response and the displacement electric field for Weyl semimetal in the frequency domain. Specifically, magnetic Weyl semimetals, characterized by broken time-reversal symmetry, can induce the anomalous Hall effect and support nonreciprocal surface plasmon polariton modes. With different azimuthal angles of incidence, the dispersion of the intrinsic non-reciprocal surface modes are quiet asymmetry with respect to the wavevector. When the twist angle equals to 0°, the SPP modes from the two surfaces match each other for the whole azimuthal angles, indicating the perfect coupling between them. However, with the increased twist angles, the arisen mismatches in return damage the coupling. In this way, the intrinsic non-reciprocal nature of these surface modes allows for effective control of heat transfer through the relative rotation of parallel slabs, eliminating the need for surface structuring or external fields.

In addition to near-field heat transfer between two surfaces, the physical mechanism between two rotating nanostructures [878] are also of great interest. Compared to the situation in the absence of rotation, the radiative heat transfer between two rotated nanostructures can be increased, decreased, or even reversed. This phenomenon reveals the unintuitive effects arising from the simultaneous transfer of energy and angular momentum in pairs of rotating nanostructures. Specifically, the sign of the torque is determined by the difference in their rotation frequencies, becoming zero when the rotations are synchronized. In contrast, the temperature of the nanostructures affects only the magnitude of the torque. The behavior of the power transferred is more complex, both its sign and magnitude result from a non-trivial interplay between the rotation frequencies and temperatures. The rotation of the nanostructures facilitates energy transfer, akin to the effect of a temperature difference between them. In the regime where thermal and rotational frequencies are significantly lower than the resonant frequencies of nanostructures, novel energy transfer behaviors emerge. These behaviors can be effectively applied to any material structure exhibiting a dipolar resonance, including large molecules. This application opens up new possibilities for controlling radiative heat transfer between nanoscale objects.

These pattern-free materials mentioned above provide the possibilities to design efficient thermal modulators without relying on nanofabrication techniques, and paves the way to apply natural materials in manipulating heat flux in an active way.

16.2.2 Metamaterials

Metamaterials and surface plasmon have been rapidly developed in the past ten years. Due to the special electromagnetic properties and strong electromagnetic field engineering ability, metamaterials have great potential application in near-field thermal radiation regulation. Combined with the localized characteristic of 2D materials, such as graphene, hBN etc. in thermal radiation band, the application of metamaterials will bring great progress in the study of thermal radiation. In this section, we will give a brief description about the near-field thermal modulation between metamaterials including 3D and 2D cases.

3D metamaterials are always related to hyperbolic materials in NFRHT studies [866, 867], which can significantly enhance the heat transfer due to the novel hyperbolic characteristic. Among metamaterials, gratings are simple structures to design, which are commonly used to improve the near-field heat flux [879]. Similar to anisotropic natural bulk materials, structured gratings can also induce in in-plane optical axis through appropriate period and filling factors selection, enabling the possibility to manually design the distribution of anisotropic permittivity [880]. Systems featuring angle-sensitive heat transfer between two misaligned polar/metallic gratings have been demonstrated to function effectively even at low filling factors. This capability allows for the manipulation of heat flux at the nanoscale, which can be beneficial for thermal management and utilization in microelectromechanical and nanoelectromechanical devices.

Magic-angle twisted bilayer systems have been shown to produce a range of exotic phenomena in 2D electronic and photonic platforms. The twisted photonic hyperbolic metasurface inspired by the concept of magic-angle offers a novel approach to breaking geometric symmetry and inducing topological transitions in isofrequency dispersion to control the photon carrier density. This paper will introduce the application of 2D metamaterials in near-field modulation.

Specifically, using twisted hyperbolic systems composed of bilayer graphene gratings, the topological transition of plasmon polaritons in NFRHT can be achieved [881]. The topology of hyperbolic SPPs in the twisted system can be adjusted from open (hyperbolic) to closed (elliptical) contours at a specific photonic transitional angle, effectively modulating radiative heat transfer. The underlying physics are qualitatively interpreted by analyzing the dispersion of individual metasurfaces in wave-vector space, as well as the unique anti-crossing features associated with the topological nature of the number of anticrossing points. This analysis clearly highlights the significant role played by the rotation angle.

The application of graphene gratings on isotropic slabs offers the potential for high-precision nanoscale thermal management and agile thermal switching [882]. This is achieved through the coupling of the substrate material with the graphene gratings, resulting in the formation of various modes, including graphene hyperbolic SPP modes, the coupled SPhP-SPP modes, and the coupled elliptic modes. These modes are influenced by the degree of alignment between the two interfaces, which is determined by the twist angle. The coating-twisting system is characterized by its strong practicality, ease of implementation, innovative construction, and simple structure, making it a promising approach for achieving precise thermal management and switching at the nanoscale. Similarly, a twisted thermophotovoltaic system incorporating hBN gratings can attain a remarkably high energy efficiency, reaching nearly 53% of the Carnot efficiency [883]. Additionally, the system can deliver an output power of up to 1.1\times10⁴ \mathrm{W}\cdot {\mathrm{m}}^{-2} without maintaining a significant temperature gradient. This demonstrates the potential of twisted near-field thermophotovoltaics, paving the way for tunable, high-performance thermophotovoltaic systems and advanced infrared detection technologies. Nonetheless, while mechanically driven grating structures may pose challenges for efficient active control of transmission properties, similar effects could be achieved using materials such as liquid crystals or metal ferromagnetic structures, where the optical axis can be readily manipulated through external fields.

Multilayer structures, in addition to gratings, have emerged as a flexible solution for NFRHT modulation due to the additional twisting degree of freedom between the inter-layers of the emitter and receiver. A commonly employed structure is the black phosphorus (BP)/vacuum multilayer [887], which supports multiple anisotropic surface plasmon polaritons (MASPPs) that enhance photon tunneling. In multilayer BP systems, the combination of integral twist and interlayer twist can generate an extremely asymmetric photonic transmission mode. Specifically, the integral twist plays a crucial role in modulating NFRHT for the BP/vacuum multilayer system. Due to the mismatch of MASPPs, the modulation ratio can reach up to 40% by varying the integral twisted angle between two bodies. Moreover, the inter-layer twist introduces an additional degree of freedom for actively controlling NFRHT, achieving a further modulation ratio of 13.6%. The insights gained from this study provide a robust approach for modulating NFRHT between multilayer metamaterials, facilitating advancements in energy conversion and thermal management.

Especially, among multilayer structures, bilayer systems represent a special case due to their ability to form a magic angle [885]. Twisted bilayer graphene (TBLG) has recently gained attention as a versatile platform for investigating various exotic transport phenomena. When the twist angle approaches the magic angle, TBLG exhibits a significant reduction in Drude weight, which can lead to over 10,000-fold suppression of heat flux. This effect is particularly pronounced under conditions of sufficiently low temperature, a tailored chemical potential, and a small carrier-scattering rate. This work underscores the immense potential of magic-angle TBLG in controlling thermal transport.

16.2.3 External field controlled structures

To gain a higher degree of regulation degree, in addition to the patterned solution, applying external field is also an efficient way. Usually, applied voltage and magnetic field are widely used method, which are also compatible in the near-field thermal modulation.

Among them, the near-field thermal switch of ferromagnetic insulator based on nonreciprocal surface magnon−polaritons is a typical pattern-free application of rotation control [886], which hosts a strong twist-induced NFRHT in the presence of twisted magnetic fields. In conditions of large damping, a significant twist-induced thermal switch ratio is observed, while in small damping conditions, nonmonotonic twist manipulation of heat transfer occurs. This behavior is associated with the distinct twist-induced effects of nonreciprocal elliptic surface magnon−polaritons, hyperbolic surface magnon−polaritons, and twist-nonresonant surface magnon−polaritons. Furthermore, in ultra-small damping conditions, the near-field radiative heat transfer can be substantially enhanced by twist-nonresonant surface magnon−polaritons. This twist-induced effect is also applicable to other types of anisotropic slabs that exhibit timereversal symmetry breaking.

Additionally, the application of an external magnetic field allows for modulation of the positions and intensities of these modes, thereby enabling the tuning of NFRHT for both parallel and twisted graphene gratings [887]. This provides a greater flexibility in controlling thermal transfer. Moreover, the modulation capability of twisting can be adjusted by the magnetic field at various twisted angles, offering an enhanced degree of control over the thermal transfer process.

The application of an external magnetic field has proven efficient for regulating graphene gratings. However, applying voltage is also a viable method [888]. This suggests that current-driven graphene metasurfaces can achieve a heating-cooling transition accompanied by rotation regulation, attributing to the nonreciprocal photon occupation number in the active graphene metasurface and the hyperbolic anisotropy inherent in the passive graphene metasurface. Band splitting and degeneracy are observed at resonant frequencies when subjected to rotation regulation. This indicates that two types of photonic topological transitions at a specific thermal magic angle for heating and cooling modes exist. These transitions result from the interplay between the nonreciprocal and non-local properties of the graphene metasurface. This discovery opens new avenues for nanoscale energy transport and thermal management, leveraging nonreciprocity.

16.3 Future outlook

Here we will give the potential development trends including theory and experiment challenges, and the promising avenues for applications.

16.3.1 Theory

The existing theoretical calculation methods for NFRHT twist modulation mainly rely on the effective medium theory (EMT) [880], scattering matrix formalism [890] and rigorous coupled wave analysis (RCWA) [891, 892].The EMT is an easy and high efficient method, but its application is limited, only when the period of the structure is much smaller than the wavelength of thermal radiation, and the near-field spacing is also limited. For periodic patterns, the precise solution based on scattering matrix formalism and RCWA is more universal. The accuracy of computing depends on the number of reciprocal lattice vectors used in the Fourier transform. However, to guarantee convergence, the memory and time for computing will significantly increase, putting forward great demanding on the computational resources. In the future, an effective and universal calculating method will be of great significance.

16.3.2 Experiment

In recent years, the experiments of NFRHT have made great development, and the experiments of variable spacing systems [893] based on traditional photoresist spacers have greatly accelerated the process, which facilitates the study of NFRHT. However, experiments on NFRHT especially for rotation control have not been reported. The lack of experiments has affected the further development of rotational regulation. Therefore, further development of variable spacing experiments with precise controllability in the future will greatly promote the research of rotation regulation.

16.3.3 Applications

The twist-induced NFRHT modulation has several typical applications, including thermal management [868], thermal switches [886], thermal transistors [894], and thermal photovoltaic devices [883, 895]. In the near field, these devices exhibit significantly enhanced performance compared to the far-field scenario. Concurrently, the continuous advancement of Moore’s Law has led to increased demand for computing power in emerging markets such as 5th Generation Mobile Communication Technology, artificial intelligence, and automotive electronics. As chip integration and miniaturization progress, chip functionality and performance have improved, but this has also resulted in increased heat output, exacerbating heat dissipation challenges. Consequently, NFRHT modulation presents a promising and efficient heat transfer solution, potentially becoming a focal point for the industry in the near future.

16.4 Summary

Twist-controlled NFRHT can be achieved by non-structural materials with in-plane anisotropy, or by structured surfaces possessing surface anisotropy flexibly. On this basis, the applied external voltage or magnetic field can further increase the modulation freedom. As an active modulation method, rotation control is helpful to realize the goal of non-contact, energy saving and environmental protection. However, the NFRHT calculation based on patterned structures are limited. A fast and accurate calculation method needs to be proposed. Besides, the lack of experiments further hamper the theoretical study of rotational regulation.

17 Review of the photon-enhanced thermionic emission energy converter

Xinqiao Lin¹, Ousi Pan¹, Zhimin Yang², Yanchao Zhang³, Jincan Chen¹, Shanhe Su^1,*

　 ¹Department of Physics, Xiamen University, Xiamen 361005, China

　 ²School of Physics and Electronic Information, Yan’an University, Yan’an 716000, China

　 ³School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China

17.1 Background

Solar energy is universally recognized as a highly promising path towards carbon neutrality, considering its status as an inexhaustible, eco-friendly, and omnipresent energy resource [896]. Over the past few decades, advancements in this renewable energy source have sparked considerable interest in high-efficiency solar cell technologies [897–899]. In recent years, thermionic emission, a process that directly converts a portion of heat energy into electricity (contrasting with photovoltaics, which rely on the photovoltaic effect), has undergone significant development [900–902]. Currently, its emission of current follows the Richardson−Dushman law [903], theoretically enabling conversion efficiencies exceeding 30% in thermionic energy converters [904]. In 2010, Schwede et al. [905] introduced a novel concept: photon-enhanced thermionic emission (PETE), which integrates both photoelectric and thermionic effects to facilitate photoelectric energy conversion under high-temperature conditions. This has garnered considerable attention among theoretical models of various emerging photoelectric converter technologies [906, 907].

17.2 Past to current development

17.2.1 Theory

Basic PETE model. In theoretical calculations and simulations of the PETE converters, certain researchers [905, 908, 909] often simplify the model by disregarding the spatial distribution of carriers within the semiconductor cathode. They adopt a zero-dimensional (0-D) model, which neglects spatial variations of the relevant variables. Within this framework, PETE can be analyzed using an ideal particle balance model akin to that employed for photovoltaic cells. The net current density of the PETE converter can be formulated as the difference between the total photo-generation rate of electrons, G, and the non-equilibrium recombination rate, R:

(64)G-R={\displaystyle \frac{J_C-J_A}{qL}},

where q represents the electron charge, L denotes the electrode gap width, and J_C and J_A are the emission current densities at the cathode and anode, respectively.

For a semiconductor cathode that integrates both thermionic emission and photoelectric effects, it has been observed that the cathode current density J_C is directly proportional to the electron concentration n at the cathode’s conduction band [905]. This relationship can be expressed as

(65)J_C=q n\sqrt{{\displaystyle \frac{qk{T}_C}{2\pi m_e}}}{\mathrm{e}}^{-\frac{\chi +E_{Cm}}{k{T}_C}},

where m_e signifies the effective mass of the electron, T_C denotes the cathode temperature, \chi represents the electron affinity, and E_{Cm} is the energy difference between the cathode vacuum energy level and the maximum potential barrier {\psi }_m in the inter-electrode space (which equals zero when space charge effects are disregarded).

In addition, the reverse emission current density at the anode, J_A, adheres to the standard thermionic emission formula [903]:

(66)J_A=A T_A^2\exp \left(-{\displaystyle \frac{E_{Am}}{k{T}_A}}\right),

where A=120\mathrm{A}\cdot \mathrm{c}{\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-2} is the Richardson–Dushman constant, and the other symbols retain their previous meanings, analogous to the equation for J_C.

By combining the emission, generation, and recombination terms, we can determine the electron concentration n, which subsequently allows for an assessment of the PETE converter’s efficiency \eta. Notably, when the output voltage equals the difference between the work functions of the cathode and the anode, this voltage is termed the flat-band voltage V_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}. Typically, the maximum power point (MPP) voltage V_{mpp} is close to this flat-band voltage. Furthermore, if the cathode is a p-type semiconductor with a low Fermi level, the device has the capability to produce a higher output voltage.

Furthermore, the temperature of the semiconductor cathode, T_C, plays a crucial role in determining the device’s performance. By integrating the elements that can absorb sub-bandgap radiation into the cathode, the cathode temperature T_C can be elevated, thereby augmenting thermionic emission. However, this process necessitates optimization to minimize black-body radiation losses. Lastly, the overall conversion efficiency of the PETE converter is also influenced by the efficiency of heat recovery at the anode. An optimal secondary-stage cycle can attain an output that corresponds to the Carnot efficiency, based on the anode temperature, converting anode waste heat into electrical energy and thereby boosting the overall efficiency of the PETE converter.

1D model. The thickness of the cathode, which is a key parameter in PETE devices, holds significant importance for the device’s performance. If the cathode thickness is excessively large, the PETE mode may degrade into a pure thermoelectronic mode, resulting in the loss of its unique advantages. To ascertain the optimal cathode thickness, a “1D model” can be employed to meticulously evaluate the balance between photon absorption and recombination losses.

In 2012, Varpula et al. [910] introduced a “1D model” for the cathode, utilizing numerical and semi-analytical methods, to compute the efficiency of a PETE device with silicon (Si) as the cathode. This model took into account electron diffusion within the cathode, non-uniform photon generation, and the mechanisms of volume and surface recombination within the PETE device. The model predicted that surface recombination could reduce the efficiency to below 10% at a cathode temperature of 800 K and a solar concentration of 1000, whereas at high injection levels, the efficiency could increase to 15%. In 2015, Varpula et al. [911] further developed the previous cathode “1D model” [910] by incorporating a purely numerical solution within the semi-analytical framework, making it effective even at high injection levels. They found that gallium arsenide (GaAs) and indium phosphide (InP) exhibited higher efficiencies (20%−25%) compared to silicon (10%−15%), making them ideal candidates for cathodes in PETE devices. Subsequently, in 2024, Li et al. [914] developed a zero and one-dimensional model for the PETE cathode, corrected the previously erroneous “photon−electron conservation relationship” outlined in Ref. [914], and found that, at a solar concentration ratio of 50, a silicon cathode PETE device would preferably operate in the photon-enhanced state. However, due to silicon’s poor photon absorption capabilities, the maximum photoelectric conversion efficiency was less than 4.5%.

After refining the fundamental diffusion−emission model, further explorations have been conducted using this model, encompassing the coupling of multiple physical fields and the assessment of novel cathode materials, and so on. In 2022, Qiu et al. [912] incorporated the one-dimensional steady-state continuity equation for electrons and holes into their study and simultaneously conducted light, thermal, and electrical modeling of PETE devices. They discovered that, with optimal electron affinity, the conversion efficiency could be maintained at 20% when the solar light concentration ranged from 100 to 500. In 2023, Wang et al. [913] identified InGaN as an ideal candidate for PETE converter cathodes due to its high light absorption coefficient and exceptional stability at elevated temperatures.

These studies highlight that relying solely on a “0D” model to describe PETE device cathodes is insufficient in capturing the intricate physical mechanisms of these devices in practical applications. To gain a deeper understanding of the underlying physical properties of PETE devices, it was imperative to upgrade the cathode model from “0D” to “1D.”

Inter-electrode model. The inter-electrode model can be categorized based on the size of the inter-electrode gap.

At the micron scale, one of the significant challenges in developing PETE devices is the space charge effect occurring between the electrodes. This effect leads to the accumulation of electrons in the electrode gap, creating a barrier that hinders the emission current of the cathode and significantly reduces the device’s efficiency. In 2014, Su et al. [915] were the first to investigate the space charge effect in PETE devices. They proposed that PETE performance is highly sensitive to space charges when the cathode emits a high current density, particularly when the gap between the electrodes exceeds a few microns. In 2015, Segev et al. [916] hypothesized that the cathode could recycle electrons that fail to cross the barrier. By integrating this electron recovery effect with the space charge effect, they found that the negative space charge loss was lower than previously reported, enabling PETE devices to have larger electrode gaps. Additionally, positive ion neutralization is another effective strategy to mitigate the impact of negative space charge. In 2012, Ito and Cappelli [917] demonstrated that positive ions for neutralizing negative space charges can be generated when electrons collide with resonantly excited cesium atoms in PETE devices. More recently, in 2024, Wang et al. [918] discovered that cesium atoms can be excited by resonant photons from sunlight to achieve associative ionization, and under these conditions, the optimal electrode gap range can exceed 5 μm.

At the sub-micron scale, when the electrode gap is reduced to a size equal to or smaller than the characteristic wavelength of thermal radiation, as defined by Wien’s displacement law, the Stefan−Boltzmann law no longer applies, and near-field effects become crucial [919]. In this scenario, both the thermal radiation effect and the space charge effect become significant factors in optimizing the electrode gap of TEC devices. In 2019, Wang et al. [920] were the first to consider both the space charge effect and the near-field thermal radiation effect in PETE (photoemission thermionic energy conversion) devices, finding that the optimal inter-electrode gap range was 0.5−2 μm at solar concentrations of 100−1000 times. In 2021, Rahman et al. [907] emphasized that, under high illumination concentrations, PETE devices utilizing Si or GaAs as cathode materials can only exhibit the “PETE mode” in sub-micron scale inter-electrode gaps, where the near-field thermal radiation effect plays a significant role.

At the nanoscale, unique phenomena such as the quantum tunneling effect and the image force effect of electrons become significantly observable. In 2016, Wang et al. [921] were the first to consider both the quantum tunneling effect and the image force effect in PETE devices, revealing their impact on properties such as the energy barrier and demonstrating that a nanogap can substantially enhance PETE performance. In 2023, Wang et al. [922] combined the photon tunneling effect with the electron tunneling effect and discovered that a conversion efficiency of 34.7% is achievable at a cathode temperature of 472.5 K, offering a promising strategy for the design of highly efficient thermionic emission devices that operate at lower temperatures.

17.2.2 Experiment

Structure configuration. In terms of structural configuration, PETE devices can be categorized based on the location of light incidence: transmission type (light incident from the cathode), reflection type (light incident from the anode), and side-irradiation type (light incident from the side) [923]. Most PETE converters adopt the transmission configuration due to its intuitive design. However, the reflective configuration has gained popularity as it removes the limitation of the cathode substrate, allowing for more diverse cathode substitutions. This configuration necessitates the use of a conductive material with high light transmittance for the anode. For instance, in 2023, Xie et al. [924] employed a reflective structure utilizing ITO (indium tin oxide) conductive glass with high light transmittance as the anode. Additionally, the side-illuminated configuration is typically used in high-voltage devices comprising multiple PETE electrode pairs (cathode−anode) connected in series [925]. In this configuration, each electrode pair receives light, with each anode serving as an electrical contact to the subsequent cathode. Operating at the same temperature, this series connection does not require additional area or increase the ohmic resistance of the device, unlike a top-down interconnection.

Spacer fabrication. In terms of spacer fabrication for PETE devices with micro or nano inter-electrode gaps, several challenges persist in production technology. In 2019, Nicaise et al. [926] proposed that the establishment of micro and nano gaps in experiments relies primarily on spacers between electrodes, which must possess heat insulation, high temperature resistance, and mechanical robustness. In 2021, Bellucci et al. [927] achieved the thinnest record of 0.3−3 μm by depositing patterned ceramic films of alumina ({\mathrm{A}\mathrm{l}}_2{\mathrm{O}}_3) and zirconia ({\mathrm{Z}\mathrm{r}\mathrm{O}}_2). In 2023, Xie [928] utilized a 150-micrometer-thick glass sheet sandwiched between the cathode and anode as a partition, and stacked additional glass sheets to incrementally control the spacing between the anode and cathode. To meet testing requirements under high temperature conditions, ceramic heating elements were also incorporated into the performance test system. In 2023, Zhang et al. [929] investigated the impact of electrode spacing structure on the optimal performance and gap size of TEC devices. They found that, under a solar concentration ratio of 100, the maximum efficiency of devices with a spacer layer (5.3%) was significantly lower than that of devices without a spacer layer (8.7%) at different optimal gaps, due to the parasitic thermal conductivity of the spacer layer. For devices without spacers, the optimal gap ranged from a few tenths of a micron to a few microns, whereas for devices with spacers, the optimal gap was approximately 8 μm. Additionally, the issue of electrical short circuits caused by thermal expansion in sub-micron inter-electrode gaps requires further investigation.

Environmental optimization. As previously mentioned, cation neutralization can effectively address the issue of irreversible losses associated with the space charge effect [917, 918]. To mitigate the space charge effect and enhance output, it is advisable to introduce positive ions into the gap after vacuuming the chamber. In Wang’s experiment [930], a cesium glass bubble, an oxygen-free cesium copper storage tube, and a heating tube were employed as the source for thermally evaporating cesium. The cesium glass bubble contains several grams of cesium metal. During operation, the bubble is inserted into the cesium storage tube, and the oxygen-free copper tube is compressed using hydraulic pliers to shatter the bubble inside, enabling the liquid cesium metal to flow into the tube. Subsequently, the heating sleeve is attached to the exterior of the storage tube and heated, causing the cesium metal to vaporize and disperse into the vacuum chamber. Both the chamber and heating sleeve temperatures are regulated by a temperature control system. Upon complete evaporation of the cesium within the bubble into the chamber, it is estimated that a cesium vapor pressure of at least 10 Pa can be achieved, which is sufficient for conducting experiments on space charge neutralization.

17.3 Future outlook

17.3.1 High dimensional model

Firstly, despite the establishment of the fundamental physical model for PETE, there is a need to enhance the dimensionality of these models. It is well-recognized that cathodes are inherently three-dimensional objects, and for an accurate assessment of device performance, it is crucial to elevate various cathode physical models to higher dimensions. For instance, cathodes with non-uniform work functions may exhibit distinct properties in different regions [931]. Specifically, when a Fresnel lens is used to concentrate sunlight onto a PETE device, if all light rays converge at a central point on the cathode surface, the temperature at the cathode’s center will be higher than its surroundings, leading to varied thermionic emission effects across different regions [932]. Additionally, the intensity of light may vary across regions, resulting in differing rates of the photoelectric effect and consequently different concentrations of photo-electrons and carriers. Furthermore, in the inter-electrode space behind the cathode surface, the space charge effect is also expected to be more accurately represented in three dimensions, as demonstrated in the work by Raja [933]. However, they utilized a 3D finite element modeling method, which requires significant computational resources. If analytical formulas for 3D models could be derived, the computational efficiency would be greatly enhanced.

17.3.2 New electrode material

Secondly, targeting nanowire materials for cathodes may lead to higher conversion efficiency. Traditional thin film cathodes reflect a significant amount of sunlight, and photogenerated electrons have to traverse a long distance (several microns or more) to reach the emission surface. Fortunately, the nanowire (NW) offer a long optical path and a short electron transport distance, enabling full absorption of incident light while electrons only need to move a nanometer distance to reach the emission surface. The ordered structure of the NW arrays has also been proven to effectively increase the length of the optical transmission path. Currently, the role of NWs in enhancing light absorption and electron emission has been thoroughly validated. For instance, in 2019, Liu et al. [934] introduced a GaAs nanowire cathode with exponential doping and gradient Al components for PETE devices, deducing an optimal nanowire length of 300−340 nm based on a theoretical model. In 2023, Xie et al. [924] employed the GaAs nanowire array as the cathode for a reflective PETE device, calculating the effects of the period, duty cycle, and height of the NW array on absorbance and conversion efficiency using the three-dimensional finite-difference time-domain method. Furthermore, in 2024, Diao and Xia [935] considered axial and radial diffusion and emission behaviors of electrons on nanowires simultaneously, establishing a quantum efficiency model for PETE devices with single GaAs nanowires and nanowire array cathodes based on the two-dimensional continuity equation, verifying the superiority of nanowire cathodes over thin film cathodes in photoemission capability.

17.3.3 Hybrid energy system

Furthermore, many researchers are actively exploring the integration of PETE devices with other types of devices. In 2019, Datas and Vaillon [936] suggested that the power generation performance of thermionic photovoltaic (TIPV) hybrid converters could be further enhanced at the near-field scale. In 2022, Qiu et al. [912] developed a numerical model for a PETE-Stirling hybrid cycle system, comprehensively considering optics, electricity, and thermodynamics. This system achieved an output power density of 162.65 kW/m² and a conversion efficiency of 32.8%. In 2023, Qiu et al. [937] demonstrated that attaching photovoltaic cells could increase the conversion efficiency of PETE devices by 4%−8%, although a portion of thermal radiation energy still couldn’t be converted into electricity through the photovoltaic effect. Additionally, Zheng et al. [938] introduced an isothermal thermal radiation enhanced PETE (iTR-PETE) structure in 2023. This structure utilizes a P-type cathode to absorb high-energy photons and an n-p junction anode to convert low-energy photons for thermal radiation. Compared to traditional systems, this coupled system exhibited a 92.3% improvement in conversion efficiency, showcasing its advantages and potential.

17.4 Summary

In summary, this article presents a comprehensive review of the development and future outlook of PETE converters. First, we delve into the fundamental operating principles of PETE converters from a theoretical perspective, covering the physical model of the semiconductor cathode and the inter-electrode interactions. Next, we outline key aspects of the fabrication process, including the overall device structure, the techniques for achieving flexible inter-electrode spacing, and the optimization methods for the inter-electrode environment. Finally, we discuss the future development trends. On the theoretical side, we advocate for further research into high-dimensional models to improve prediction accuracy. Experimentally, we anticipate advancements in more efficient cathode materials and emphasize the importance of integrating PETE devices with other compatible technologies to maximize their potential. Although significant challenges remain before PETE converters can be practically implemented, their higher theoretical conversion efficiency and broader application prospects in solar technology make them a promising area for ongoing exploration and research.

18 Structural design and optimization of thermal metamaterials

Qingxiang Ji, Muamer Kadic^*

　 Université de Franche-Comté, Institut FEMTO-ST, CNRS, Besançon 25000, France

18.1 Background

Transformation optics, developed in 2006, is based on the form invariance of physical governing equations under designed coordinate transformations. It has become a key tool for studying the manipulation of electromagnetic waves [939–941], and has inspired innovations such as invisible cloaks, superlenses, and more [942].

Transformation thermodynamics, the counterpart in heat transfer, focuses on controlling heat flux for new thermal functionalities like negative conductivity, thermal cloaking, and thermal camouflage [943–946]. While crucial for designing thermal metamaterials, it often results in complex, anisotropic, and inhomogeneous constitutive parameters, making real-world applications challenging.

Huang et al. addressed this by using effective medium theory to approximate the ideal conductivity profiles from transformation thermodynamics [947]. Structures made of alternating isotropic layers have since been fabricated and tested in steady-state and transient conditions [948, 949], but these multilayer designs offer limited effective thermal conductivities [950].

To expand the range of achievable conductivities, topology optimization [951, 952] has been used to generate optimal configurations from isotropic materials, overcoming anisotropy challenges but resulting in complex, hard-to-manufacture structures.

New approaches are needed to offer both broad access to effective thermal conductivities and adaptability to complex geometries. Insights from acoustics, where designed microstructures approximate continuous medium properties [953, 954], could inform future thermal metamaterials design.

In the following sections, we outline the road map for achieving target ETCs through microstructure optimization and discuss future directions for advancing the field.

18.2 Past to current development

Recalling the steady-state heat conduction equation without heat sources:

{\mathrm{\nabla }}^{\prime }\cdot ({\mathit{k}}^{\prime }{\mathrm{\nabla }}^{\prime }{T}^{\prime })= 0,

where T^{\prime } is the temperature and {\mathit{k}}^{\prime } denotes the heat conductivity. Transformation thermodynamics theory states that this equation is form-invariant under coordinate transformation if

(67)\mathit{k}={\displaystyle \frac{\mathit{J}{k}^{\prime }{\mathit{J}}^{\mathrm{T}}}{det(\mathit{J})}} .

Here, the transformed parameter \mathit{k} becomes anisotropic and space-dependent [955]. The theory posits that anisotropic properties can be approximated by carefully designed isotropic microstructures. This involves two steps: first, designing microstructures and determining their ETCs; second, optimizing the microstructures to match the target ETCs from transformation thermodynamics. The challenge is the structural design and accurate estimation of ETCs. Effective medium theory is a common method for evaluating ETCs, and alternate-layer structures are a widely used example [948, 949]. It also applies to complex geometries; for example, Ji et al. [956] designed fiber-crossing microstructures for heat concentration, and Pomot et al. [957] achieved acoustic cloaking with perforated structures.

Microstructures should be optimized to achieve the required ETCs. Different perforations have been proposed for thermal cloaking, showing broad applicability [958]. The effective medium theory is useful but becomes complex for intricate geometries. In such cases, two-scale homogenization theory is employed to determine equivalent thermal properties with small perturbations [959, 960].

Considering a periodic medium with cubic elementary cells [0,\eta {]}^3 with side length \eta =1/N, where N is a large integer and \eta \ll 1, the equivalent property of the periodic medium is

{\mathit{k}}_h=\left(\begin{array}{ccc}⟨k⟩-⟨k{\mathrm{\partial }}_x{V}_1⟩& -⟨k{\mathrm{\partial }}_x{V}_2⟩& -⟨k{\mathrm{\partial }}_x{V}_3⟩\\ -⟨k{\mathrm{\partial }}_y{V}_1⟩& ⟨k⟩-⟨k{\mathrm{\partial }}_y{V}_2⟩& -⟨k{\mathrm{\partial }}_y{V}_3⟩\\ -⟨k{\mathrm{\partial }}_z{V}_1⟩& -⟨k{\mathrm{\partial }}_z{V}_2⟩& ⟨k⟩-⟨k{\mathrm{\partial }}_z{V}_3⟩\end{array}\right),

where {\mathrm{\partial }}_x:= \mathrm{\partial }/\mathrm{\partial }x, {\mathrm{\partial }}_y :=\mathrm{\partial }/\mathrm{\partial }y, {\mathrm{\partial }}_z:=\mathrm{\partial }/\mathrm{\partial }z, and ⟨\cdot ⟩ is the mean operator. V_1(x,y,z), V_2(x, y,z), and V_3(x,y,z) are obtained by solving the following auxiliary problems [960]:

(68)\begin{array}{rl}& \mathrm{\nabla }\cdot \left[k(x,y,z)\mathrm{\nabla }(V_1-x)\right]=0,\\ & \mathrm{\nabla }\cdot \left[k(x,y,z)\mathrm{\nabla }(V_2-y)\right]=0,\\ & \mathrm{\nabla }\cdot \left[k(x,y,z)\mathrm{\nabla }(V_3-z)\right]=0.\end{array}

This method is particularly effective for complex geometries, where effective medium theory becomes cumbersome.

Once ETCs are determined, the next step is optimizing the thermal metamaterials. This involves solving an inverse problem: determining optimal microstructures to achieve desired ETCs. Traditional methods like particle swarm optimization [961] and topology optimization [962] have been used to obtain optimal designs, while numerical methods such as genetic algorithms and artificial bee colony algorithms [963] are commonly employed for microstructure optimization.

Despite progress with artificial microstructures, challenges like low computational efficiency, local minima, and convergence remain. Target thermal conductivities depend on factors like background conductivity, geometry, and desired functionality, requiring repeated optimizations across the device. Deep learning is proposed as a solution to these issues. The process involves:

i) Establishing an initial database using homogenization or finite element analysis.

ii) Training a deep learning model to predict ETCs from geometry parameters.

iii) Generating extended databases with reduced computational cost.

iv) Building an inverse neural network to determine geometry from desired ETCs.

Although inverse networks may face convergence issues due to one-to-many mappings, probabilistic design spaces can help. For example, a 1% probabilistic design space contains samples with ETCs deviating less than 1% from target values, allowing the selection of manufacturable microstructures [964–966].

18.2.1 Application

The above theory provides a fundamental tool to determine the thermal conductivity properties required by transformation thermotics. With this framework for thermal metamaterials design, the next focus is on thermal functionality and novel devices for practical applications.

Since all the desired ETCs for given materials can be robustly achieved by designing and optimizing microstructures, the strict material−property requirements of thermal metamaterials are fully met. Various thermal functionalities and metadevices can be designed freely. Verified applications include extensively studied thermal management techniques such as cloaking, concentrating, rotating, and trapping [967]. Beyond these conventional applications, emerging thermal regulations such as nonlinear thermotics, Janus characteristics, nonreciprocal heat transfer, and topological effects have also been noted [968].

The development of thermal metamaterials also inspires new possibilities in thermal information processing and signal control. Implementations of these thermal metamaterials include dynamic thermal information processors, encryption systems, adaptive metasurface platforms, and thermal logic gates [969, 970]. Since thermal properties like conductivity and heat capacity play a major role in these applications, the established theory and methods are well-suited for implementation.

Beyond thermal conduction, the same theory and methods apply to other physical fields governed by similar equations, such as electricity, mass transport, and particle diffusion. The key difference is that in these fields, the optimized targets will be the equivalent physical properties, such as electrical conductivity or diffusion coefficients, rather than thermal conductivity.

18.3 Future outlook

Although progress has been made in thermal regulation by optimization of thermal metamaterials, it remains a challenge to obtain more complex and novel functionalities. In the thermal transfer field, only heat conduction is considered. Convection and radiation are also fundamental modes of heat transfer and are always accompanied by conduction, but they are seldom considered. Adding convective terms in the heat governing equations imposes much complexity on the design of thermal metamaterials, while the radiative term is not suitable for direct coordinate transformation. To conduct structural design and optimization of thermal metamaterials in the presence of all heat transfer modes remains to be explored.

Recently the works are focused on steady state conditions in the sense that heat capacity is ignored, and on static cases in the sense that thermal properties are fixed while not influenced by time, temperature and other stimulus. It is expected that thermal metamaterials coupled with external stimuli, dynamic manipulation or smart materials can achieve more powerful thermal functionalities. For these attempts, multiphysical fields should be considered, which calls for more complex multi-objective optimization.

18.4 Summary

In summary, we present a design paradigm for the structural design and optimization of thermal metamaterials that are both feasible and manufacturable in practical applications. With the advancement of 3D printing, these thermal metamaterials can be easily assembled into various types of thermal metadevices, enabling diverse heat regulation functionalities. The established theory and methods will also be applicable to future novel functionalities, as long as equivalent thermal properties are involved.

19 Topology optimization approaches for thermal cloaks/metamaterials

Garuda Fujii^*

　 Faculty of Engineering, Shinshu University, 4-17-1 Wakasato Nagano 380-8553, Japan and Energy Landscape Architectonics Brain Bank (Elab2), Interdisciplinary Cluster for Cutting Edge Research, Shinshu University, 4-17-1 Wakasato Nagano 380-8553, Japan

19.1 Background

Since the pioneering work of Bendsøe and Kikuchi [971], topology optimization has been extensively developed with a robust mathematical background and its application has expanded significantly with recent advances in computational power. From its origin within the field of elasto-mechanics in compliance minimization subject to a volume constraint, topology optimization has grown rapidly to encompass a broad spectrum of physical phenomena, leading to solutions for numerous design problems. Additionally, the diversity of structural representations for modeling configurations has expanded considerably along with the sophistication of optimization algorithms used to find optimal designs. Rapid advances in methodologies incorporating artificial intelligence and other cutting-edge techniques have also accelerated the development of topology optimization.

Metamaterials and cloaking represent particularly exciting applications of topology optimization because of the challenges involved in achieving the necessary material properties and performance. In recent years, a growing body of research has emerged reflecting their potential. Initially focused on controlling electromagnetic waves, these technologies have since been extended to manipulate various physical phenomena, with using thermal metamaterials to control heat being among the most successful research areas.

While transformation thermotics [972] has been widely adopted to determine the arrangement of anisotropic thermal metamaterials for desired thermal manipulation, the use of topology optimization to design thermal cloaks and metamaterials has gained substantial attention as an effective means of enhancing performance.

This chapter focuses on the application of topology optimization techniques to the design of thermal cloaks and metamaterials. Although comprehensive reviews on the optimal design of thermal metamaterials already exist [973, 974, 974], we emphasize here the specific aspects related to topology optimization. To assist readers in applying these techniques, a simple checklist is provided below.

19.1.1 Checklist for topology optimization

Does the topology optimization change the topology of structure during optimization? There are a few papers with “topology optimization” in the title that, in fact, discuss sizing or shape optimization. The most attractive feature of topology optimization is its high degree of design freedom, which allows not only changes to the outer shape of the structure but also alterations to its topology such as the creation of new pores during the optimization process.

In the early stages of topology optimization, certain approaches were proposed in which the topology could not be altered (i.e., these were essentially shape optimization methods), depending on factors such as the update scheme of design variables, sensitivities/topological derivatives, and the properties of the design variables. However, such approaches are rarely seen today. Therefore, to prevent misleading readers, it is crucial to verify whether the optimization method allows topological changes of structures through the definition of the design variables or the representation of the structure.

Is the optimization problem properly set up, including the objective function(s), constraint(s), and scheme? Does the objective function effectively evaluate performance? Setting the objective function(s) is the most critical element in defining the optimization problem and clearly indicates what designers aim to achieve through solving it.

Is the constraint truly necessary? Topology optimization problems are generally formulated as constrained design problems, in which the manipulation of the physical quantity of interest is often framed in terms of one or more objective functions. Thus, the governing equation(s) and boundary condition(s) that the physical quantity satisfies, are imposed as constraints. However, adding unnecessary constraints can inadvertently exclude ideal optimal solutions from the solution space. For example, a volume constraint that is not essential for achieving the performance goals of thermal metamaterials or related metadevices might limit the solutions of optimization. Unless such constraints are imposed purely for mathematical reasons or manufacturing considerations, the optimization problem becomes artificial, potentially ruling out good solutions.

How is an ill-posed problem in topology optimization addressed? Topology optimization problems are often classified as ill-posed, meaning they typically lack well-defined solutions because the original topology optimization allows for the creation of infinitesimal structures within the design configuration. It is incorrect to assume that such infinitesimal structures cannot form simply because the minimum structural scale is limited by the discretization such as a finite element or a lattice. In fact, improving computational accuracy by refining the discretization, such as reducing the finite element size, will lead to smaller allowable structural scales. This issue relates to the existence of optimal solutions within the design problem itself and is not a direct consequence of numerical discretization. The ill-posed nature of topology optimization is commonly addressed through relaxation methods such as homogenization or the density method, or by regularization techniques like imposing a geometric constraint such as a perimeter constraint.

19.2 Past to current development

Topology optimization is defined by its core components: methods of structural representation, optimization algorithms, numerical techniques for analysis, objective function(s), and constraint(s). Here, we classify studies on topology optimization for thermal cloaks and metamaterials based on design scale and summarize the relevant methodologies.

19.2.1 Topology optimization on the macroscale

The fundamental formulation for a thermal cloak operating in the steady state heat conduction is described as follows:

(69)\begin{array}{rl}& \underset{\mathit{X}}{\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}}\mathrm{\Psi }=\frac{1}{{\mathrm{\Psi }}_n}{\int }_{{\mathrm{\Omega }}_{\mathrm{o}\mathrm{u}\mathrm{t}}}|T-T_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}{|}^2\mathrm{d}\mathrm{\Omega },\\ & \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{\nabla }\cdot \left(\kappa (\mathit{x})\mathrm{\nabla }T\right)=0\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ & \mathrm{\nabla }T\cdot \mathit{n}=0\mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_{\mathrm{i}\mathrm{n}\mathrm{s}},{\mathrm{\Gamma }}_{\mathrm{o}\mathrm{b}\mathrm{s}},\\ & T=T_{\mathrm{l}\mathrm{o}\mathrm{w}}\mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_{\mathrm{l}\mathrm{o}\mathrm{w}},\\ & T=T_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}\mathrm{o}\mathrm{n}{\mathrm{\Gamma }}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}},\end{array}

where \mathit{X} is the design variable representing configurations, \mathrm{\Psi } the objective function to be infimized, {\mathrm{\Psi }}_n the normalization factor, T the cloaked temperature, T_{\text{target}} the desired temperature, {\mathrm{\Omega }}_{\text{out}} the domain over which the temperature difference is integrated, \kappa (\mathit{x}) the position-dependent thermal conductivity, \mathrm{\Omega } the entire domain of analysis, \mathit{n} the unit vector normal to boundaries, {\mathrm{\Gamma }}_{\text{ins}} represents the insulated boundary, {\mathrm{\Gamma }}_{\text{obs}} the outer boundary of an insulating obstacle, and T_{\text{low}} and T_{\text{high}} are the lower and higher temperatures imposed on {\mathrm{\Gamma }}_{\text{low}} and {\mathrm{\Gamma }}_{\text{high}}, respectively, as Dirichlet boundary conditions. The temperature in the objective function in Eq. (69) can be replaced with the heat flux \mathit{q}=-\kappa (\mathit{x})\mathrm{\nabla }T to achieve the target flux {\mathit{q}}_{\text{target}}=- \kappa (\mathit{x})\mathrm{\nabla }{T}_{\text{target}} by infimizing the norm | | \mathit{q}-{\mathit{q}}_{\text{target}}|{|}^2.

Thermal cloaks have been designed using level set-based topology optimization [976] explored via the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). This approach has also been applied to the design of a thermal carpet cloak [978]. While finite-element analysis is often employed, topology optimization using the Lattice Boltzmann method for steady state thermal conduction has also been demonstrated [980]. Before these works, the simultaneous realization of thermal cloaking and heat flux concentration was achieved using Solid Isotropic Material with Penalization (SIMP) approach [981, 982]. Shortly after, level set-based topology optimization was demonstrated for an undetectable thermal concentrator [983]. Undetectable thermal reversing has been developed not only for cloaking but also reversing heat flux thereby improving multiple objective functions [977]. Experimental demonstrations of these topology-optimized thermal metastructures has revealed their performance [977, 984]. Additional advancements include isogeometric shape optimization for thermal concentrators [985], followed shortly after by the development of isogeometric topology optimization methods [986]. The extended level set method has also been applied to the topology optimization of thermal cloaks [979] including three-dimensional designs. The generalized Benders decomposition procedure [987] was implemented for SIMP-based thermal cloaking optimization.

19.2.2 Multiscale topology optimization for thermal cloaks composed of metamaterials

The pioneering work [988] on topology optimization for thermal metamaterials aimed to create anisotropic thermal conductivity in thermally conductive particles for thermal shielding, heat flux focusing, and reversal. Almost a decade later, multi-scale design optimizations followed using homogenization methods to calculate the effective thermal conductivity by solving cell problems to control heat transfer on a macroscopic level [990]. This method has become the standard for multi-scale topology optimization, leading to the successful design of composite materials for thermal cloaking.

Other approaches minimize the volume of thermal metamaterials while matching anisotropic conductivity derived from transformation thermotics. Here, the SIMP method was employed for modeling thermal metamaterial cells (thermal functional cells) in topology optimization [989, 992]. This method has also been applied to multiscale designs, including cloaking sensors that receive omnidirectional external heat signals [993], as well as multi-point cloaks and undetectable concentrators [994]. Recent developments include B-spline-based level set methods for structural representation of thermal metamaterials, leading to the design of two- and three-dimensional thermal carpet cloaks [995]. Recently, data-driven, two-scale designs have enable the realization of multiple macroscopic capabilities such as cloaking, flux concentration, rotation, and reversal [991].

19.2.3 Topology optimization for multiphysical manipulation in cloaking

Because of the similarity of the governing equations and the boundary conditions for thermal and electrical conductions, simultaneous cloaking of heat and direct current has been achieved through topology optimization [996]. This has since been extended to bi-physical undetectable concentrators [997]. Multiscale designs that combine transformation theory with density-based topology optimization [989] have been extended to multi-physical/independent manipulation for both cloaking and concentrating [998, 999]. Additionally, efforts have been made to develop density-based topology optimization for mechanical cloaking under thermal load [1000], as well as thermal carpet cloaking that also provides mechanical load-bearing capability [1001].

19.3 Future outlook

Enhancing the manufacturability of optimal configurations is a crucial topic for the practical implementation of thermal cloaks/metamaterials. Recent developments in geometrical constraints via fictitious physical models [1002–1004] offer a promising solution to controlling the optimal configurations of these thermal cloaks/metamaterials. Discrepancies between computationally optimized structures and fabricated specimens can significantly degrade the performance of the optimized designs. Thus, developing methodologies that yield easy-to-fabricate structures, ensuring consistency between computational models and experimental results, is essential.

There is also significant room for improvement in the optimizers used to find optimal configurations. Many of their algorithms face challenges such as multimodality, interdependence of design variables, and the curse of dimensionality, leading to high computational costs and difficulties in finding robust solutions. Although topology optimization for thermal cloaks/metamaterials does not encounter these challenges as frequently as wave-related problems, various recent approaches have been proposed to tackle these issues such as data driven multi-fidelity methods using variational auto-encoders [1005] and evolutionary strategies incorporating Fourier-expanded level set methods [1006]. Additionally, a transformation thermotics-based analytical approach [1007] has been introduced to obtain near-optimal thermal conductivity distributions at very low computational cost.

Other emerging metastructures for heat manipulation such as illusion thermotics [1008] and thermal wave control [1009], also represent promising targets for topology optimization techniques.

19.4 Summary

This chapter outlined past and recent developments in topology optimization for thermal cloaks/metamaterials and provided a brief outlook for future research in the field. As these areas continue to evolve through advances in both physical and numerical/mathematical aspects, it is vital to stay informed about the latest research. Without doubt, the field has become exciting and competitive with researchers from diverse backgrounds contributing to advance the state of the art.

20 Topology optimization for thermal metamaterials

Zhaochen Wang, Run Hu^*

　 School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

20.1 Background

Thermal metamaterials have garnered significant attention due to their unique ability to manipulate heat flow in unprecedented ways [1010–1013]. Traditional materials are limited in their ability to control thermal behavior, whereas thermal metamaterials can achieve novel functionalities such as cloaking [1014, 1015], concentrating [1016], rotating [1017], and illusion [1018]. These functionalities are enabled through carefully designed microstructures and material compositions that do not exist naturally. The properties that surpass natural materials also pose great challenges in the design and manufacturing.

Topology optimization is a computational method that allows for the optimization of material distribution within a given design domain [1019]. By maximizing or minimizing specific objective functions, topology optimization can identify optimal material layouts that achieve desired properties and functionalities. Over the years, this method has evolved to encompass a wide range of fields, including mechanics [1020], electromagnetics [1021, 1022], and more recently, thermotics [1023, 1024]. Applying topology optimization for the design of thermal metamaterials has witnessed significant progress due to its powerful capability to provide solutions that are often infeasible or extremely complex to design with traditional methods. By leveraging advanced algorithms and computational resources, topology optimization can explore a vast design space, identifying optimal configurations of thermal metamaterials while adhering to practical manufacturing constraints.

20.2 Past to current development

20.2.1 Theory

Thanks to the proposal and expansion of transformation optics into other fields, the design of metamaterials has been effectively guided by theoretical frameworks [1025, 1026]. In the domain of thermal metamaterials, transformation thermotics [1027] and scattering cancellation methods [1028–1030] have become standard approaches for designing thermal metadevices with a variety of thermal functionalities. However, both methods exhibit certain limitations when applied in practice. Transformation thermotics is sufficiently flexible, catering to different materials, shapes, and thermal functionalities. Yet, the thermal conductivity resulting from coordinate transformation is typically highly anisotropic and may even contain singularities, making it extremely challenging to fabricate. In contrast, structures derived from scattering cancellation methods are more readily achievable, but their flexibility is limited, struggling to meet the demands of complex shapes and functionalities. Confronted with these challenges, researchers are sparing no effort for solutions that balance the complexity of design, manufacturing, and functionalities, aiming to maximize the immense potential of thermal metamaterials in heat flow regulation.

Researchers proposed a topology optimization design approach that directly targets specific temperature fields with different thermal functionalities [1031–1035]. This innovative approach bypasses the traditional reliance on coordinate transformations or scattering cancellation methods, offering a new pathway for the development of thermal metamaterials with various thermal functionalities. Fujii et al. [1033, 1034] designed thermal cloak and concentrator using level set method and covariance matrix adaptation evolution strategy. Sha et al. [1035] designed thermal illusion devices with a general topology optimization method. Take the scheme of topology optimization for thermal illusion devices as an example, as depicted in Fig.42. Firstly, define the geometric structure and boundary conditions. Then take the preset background temperature (BT) distributions that correspond to the desired thermal functionalities as the input target [Fig.42(a)]. Finally, optimize the distribution of material 1 and material 2 within the design region to steer the temperature field towards the target temperature field [Fig.42(b)]. Fig.42(c) shows some optimized structures for thermal illusion with different volume fractions of Material 1. It can be observed that this method readily yields a clear material distribution with distinct boundaries and is adaptable to a variety of shapes and boundary conditions. These kinds of methods utilizing the target BT are collectively referred to as BT-dependent design.

Although BT-dependent design has enabled the ability to design the structure to realize target thermal functionalities. This approach, however, relies on the knowledge of the BT, which may not always be feasible or practical, especially in dynamic environments where temperatures fluctuate or are unknown. Thus, researchers proposed the BT-independent design that does not require prior knowledge of the BT [1023]. This approach focuses on designing thermal metamaterials that can manipulate heat flow independently of the background temperature.

BT-independent design is achieved by optimizing the local thermal conductivity tensors and assembling topological functional cells (TFCs) that can guide heat flow according to the desired thermal functionalities. The design paradigm is depicted in Fig.43. The first step is to divide the design area of thermal metamaterials into several TFCs. Based on the functionality and geometrical shape of the targeted thermal metamaterials, the anisotropic thermal conductivity tensors required for each TFC are calculated with transformation thermotics. The second step is to perform topology optimization on the microstructure of each TFC, with the calculated anisotropic thermal conductivity tensors as the objective function. Ultimately, all the microstructures are assembled to construct the thermal metadevice, achieving the desired thermal functionalities. Subsequent studies have proved that these TFCs obtained through topology optimization can traverse full-parameter anisotropic space of thermal conductivity tensors, with design capabilities far exceeding layered structures [1036]. This approach offers greater flexibility and adaptability since it can maintain thermal functionalities regardless of the geometrical shape and background temperature, making them suitable for a wider range of applications, including those with varying or unknown boundary conditions. Nonetheless, BT-independent designs may be more complex to develop due to the need for sophisticated optimization algorithms and potentially more intricate material structures. Additionally, while they offer robustness against temperature variations, their performance might still be affected by other factors not accounted for in the design process, such as material degradation over time or interactions with other physical fields. Despite these challenges, BT-independent design represents a significant advancement in the field of thermal metamaterials, providing a powerful tool for thermal management in complex boundary conditions and various applications.

20.2.2 Application

With the theoretical advancement of topology optimization, thermal metamaterials with more complex and powerful thermal functionalities have also been realized gradually.

Early BT-dependent design was primarily utilized for the topology optimization of circular thermal cloaks [1031]. These devices, with symmetrical shapes, usually require optimization of only a 1/4 segment of the design region, significantly reducing the complexity. Subsequently, this method was extended to the design of various thermal functional devices, such as thermal concentrator [1017] and thermal illusion [1035]. However, these applications are typically limited to symmetrical shapes. Moreover, since these designs are driven by target temperature fields, the thermal functional devices created by this method are only effective under specific boundary conditions. Changes in the magnitude or direction of external temperature gradients could potentially render these devices ineffective. The subsequent BT-independent design approach, which is driven by thermal conductivity tensors, has addressed this limitation. It has not only enabled the design of thermal cloaks, concentrators, and rotators with arbitrary shapes but also ensured their effectiveness against omnidirectional external temperature gradients. This has greatly expanded the application range of thermal metamaterials. Consequently, topology optimization has also been applied to the design of more innovative thermal functional devices, such as thermal connectors and thermal reflectors.

Moreover, there has been growing interest in designing multiphysics metadevices with topology optimization. Taking metadevices that simultaneously regulate thermal and electric fields as an example, these devices require sophisticated design strategies that couple the thermal and electrical properties of the materials used. Thanks to the same-form Laplace governing equations in thermotics and electric, and the fact that both Fourier’s law and Ohm’s law follow the mathematical form of linear constitutive relation. Therefore, using topology optimization to design thermo-electric coupled metamaterials with the same functionality is relatively straightforward. Researchers have demonstrated metadevices that can cloak against both heat flux and electric current [1032, 1037], opening up new possibilities for multifunctional sensors [1038] and camouflage systems [1039, 1040]. However, designing thermo-electric coupled metamaterials with different functions with topology optimization is particularly challenging. For instance, thermal cloaking requires the tangential thermal conductivity to approach infinity and the radial thermal conductivity to approach zero, while electric concentration requires the tangential electrical conductivity to approach zero and the radial electrical conductivity to approach infinity. Furthermore, materials with high thermal conductivity often also have high electrical conductivity, and vice versa. These factors significantly increase the difficulty of the design. Thanks to the power of the algorithm, recently, Zhu et al. successfully designed transformation multiphysics metamaterials with thermal concentrating and electric cloaking functions using topological optimization [1041].This work demonstrates the great potential of topology optimization in the face of extremely complex thermal properties design.

20.3 Future outloook

Although topology optimization for thermal metamaterials has achieved significant advancements and has been widely applied, there remains substantial room for improvement in terms of more effective methods, a richer array of functionalities, and more effective practical applications. It suggests that this field is ripe for further innovation and development to enhance the performance and versatility of thermal metamaterials.

The method of topology optimization needs to evolve in two main directions: better algorithms and more manufacturable design. More efficient and accurate algorithms can explore larger design spaces and identify optimal configurations more rapidly. The integration of machine learning and deep learning techniques may significantly accelerate the design process by predicting the performance of different material distributions without the need for repeated finite element analyses [1042]. At the same time, considering fabrication constraints and material properties within the method is necessary. It is much more important to ensure the manufacturing with available manufacturing technologies even elegant structure rather than only designing it.

Furthermore, the innovation in functionalities may benefit from the exploration of new material systems. The development of advanced materials with tunable thermal properties, such as phase change materials [1043] and metamaterials with negative thermal conductivity [1044], could enable the design of unprecedented functionalities. Also, the integration of multi-material systems offers opportunities for achieving more sophisticated thermal behaviors. Moreover, the design of multiphysics and multifunctional thermal metamaterials remains an exciting area of research. Future research may be able to achieve the combination of any physical field and any functionality, providing an incredible degree of manipulate freedom. This will require the development of new optimization frameworks that can account for the interactions between different physical phenomena.

Finally, the transition from theoretical research into practical applications will be crucial for the widespread adoption of topology optimized thermal metamaterials. Future exploration is likely to focus on establishing a direct and reliable topology optimization paradigm that leverages the full potential of thermal metamaterials to effectively address actual thermal problems like energy-efficient building designs [1045] and intelligent thermal management [1046].

20.4 Summary

Topology optimization has emerged as a powerful tool in the design of thermal metamaterials, enabling the achievement of functionalities that were previously unattainable using traditional methods. From the early development of cloaking devices to the recent advancements in field-coupling and robustly printable thermal metamaterials, this field has witnessed significant progress over the past decade. Looking ahead, the future of topology optimization for thermal metamaterials relies on the innovation of enhanced optimization methods, novel functionalities, and the development of multiphysics and multifunctional devices. Also, with the potential for widespread practical applications, topology optimized thermal metamaterials hold promise for revolutionizing various industries and enabling new technologies that can address challenges like extreme thermal management and energy sustainability.

21 Thermal cloaking: A historical perspective and future outlook

Junyi Nangong^1,2, Kaihuai Wen^1,2, Tiancheng Han^1,2,*

　 ¹National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology, Chengdu 611731, China

　 ²Key Laboratory of Multi-spectral Absorbing Materials and Structures of Ministry of Education, University of Electronic Science and Technology, Chengdu 611731, China

21.1 Background

Over the past few decades, thermal cloaking technology has evolved from a concept found in science fiction to a significant research area. Similar to the concept of optical cloak that makes objects invisible to the human eye, this technology aims to make objects invisible to thermal detection by manipulating heat flow around them. Thanks to the development of materials science and successful experience in electromagnetic field, the first attempt at heat flow control was made in 2008 [1047]. Following the successful manipulation of heat flow, various thermal cloaks have been developed. As research continues, this technology is further expected to increasingly integrate with advanced fields such as deep learning technology, which can enhance its capabilities and enabling more sophisticated and customized applications. In the future, the combination of these technologies is expected to play a crucial role in military, aerospace, and electronic devices applications.

First, let us define a thermal invisibility cloak by outlining its two primary tasks: (i) Heat manipulation and (ii) thermal shielding. The first task means the cloak should control and redirect heat flow around an object, while the other ensures that the cloaking region remains insulated from external temperature changes. Inspired by the realization of invisibility cloaks at microwave frequencies [1048], the transformation theory becomes a bridge between the invisibility cloak functions and the material properties [1049]. Then a variety of thermal cloaks have been developed, including the original TO-based cloak [1050] and multilayered cloak [1051], which serve as early examples. Besides, leveraging the design principles applied to static magnetic fields [1052], another technique called scattering cancellation [1053] aids in achieving bilayer cloak in both 2D and 3D scenarios. In addition to the two primary design theories described above, a series of supplementary theories have also been introduced as essential tools for achieving the design of thermal cloaks. Subsequently, thermal cloaking has advanced not only in shaped design but also in the study of the three modes of heat transfer. In recent years, with the advancement of computer science, the design of thermal cloaks has also begun to explore integration with optimization algorithms. In this review, we will overview the historical development and current state of thermal cloaking technologies, and provides a forecast of potential future developments in thermal cloaking technologies. Fig.44 outlines the framework for thermal cloaking technology.

21.2 Evolutionary development

The theoretical development in this field must begin with the introduction of transformation theory, which can be effectively utilized as long as the governing equations of physical fields remain form-invariant under coordinate transformations. It was initially used to manipulate electromagnetic waves, and was later widely adapted to thermal domains and becomes known as transformation thermotics. It is used to design thermal cloaks by mathematically reshaping space to control how heat flows, rendering objects thermally invisible. This process includes creating materials with specifically tailored thermal properties to guide heat around the cloaked object [1049], which is the process of designing metamaterials. When designing a thermal cloak, the two most important thermal parameters are (i) thermal conductivity and (ii) the product of density and heat capacity. Taking these factors into account, the problem of designing a thermal cloak based on transformation thermotics essentially becomes the task of fabricating materials with properties that satisfy the corresponding coordinate transformations.

On the basis of transformation thermotics, manipulation of heat flux was first studied in 2008 [1047, 1062]. In 2012, Guenneau et al. [1049] made the first attempt to extended TO theory to transformation thermotics for controlling thermal conduction, and the transient thermal cloaks were realized in the following year [1050, 1063]. This thermal cloak with inhomogeneous and anisotropic thermal conductivities is called TO-based cloak, where a hole-drilling process is applied [1050] or several homogeneous sub-layers are integrated for different thermal conductivities [1063]. Besides, the design of a steady-state thermal cloak only requires thermal conductivity, leading to a multilayered cloak with homogeneous and anisotropic conductivities [1051, 1064]. In addition to transformation theory, another approach to achieve a thermal cloak is scattering cancellation. This design strategy is directly inspired by magnetostatic cloaks [1052] and has led to the realization of bilayer thermal cloaks in both two-dimensional [1065] and three-dimensional [1066] forms. Different from transformation methods that typically require inhomogeneity and anisotropy conductivities, bilayer thermal cloaks can be constructed with only bulk isotropic materials. The critical step of this method involves solving the heat conduction equation, and this process could be notably simplified if the boundary conditions are geometrically symmetrical [1067].

As mentioned, materials with anisotropic and nonhomogeneous properties are often required in transformation theory. However, it is rarely possible to find natural materials that meet these specific design requirements. Therefore, the effective medium theory (EMT) is introduced to handle these problems. Based on EMT, unconventional thermal conductivities can be achieved through the use of repeated unit structures made from isotropic and homogeneous materials, and the unit structures are typically layered in most cases [1050, 1051, 1059]. This theory is occasionally used in conjunction with the Maxwell−Garnett formula (effective-medium formula) in hole-drilling process, which aims to achieve targeted thermal conductivities, as demonstrated in Refs. [1050, 1059]. Additionally, an alternative rotation process is introduced to transform conductivity tensors into a diagonal form [1068], which enhances the applicability of EMT. The design of thermal cloaks has also extended to encompass other physical fields, resulting in the development of multiphysical or multifunctional cloak. In 2010, the concept of a bifunctional cloak was firstly introduced, capable of manipulating thermal and electric fields simultaneously [1054]. Subsequently, the electric-thermal bifunctional cloak was experimentally realized through the scattering cancellation method in 2014 [1069], successfully achieving both electric current and heat flux cloaking. After thermoelectric transport was initially established as a model for multiphysical cloaking [1070, 1071], recent advancements in the simultaneous manipulation of thermal and electromagnetic cloaking have sparked a renewed wave of research in the field of multiphysical cloaks [1072].

The invisibility cloaks discussed above typically exhibit similar shapes, either circular or elliptical when projected onto a 2D plane. This fact has inspired researchers to explore the expansion of thermal cloak shapes as a potential area for further study. In 2008, a theoretical design for a two-dimensional electromagnetic cloak featuring polygonal and arbitrary shapes was developed through the application of coordinate transformations [1073], thereby demonstrating the efficacy of transformation theory in the design of cloaks with diverse shapes. Then in 2017, the design of N-sided polygonal transient thermal cloaks was developed through the coordinate transformation [1055]. Concurrently, the concept of partitioning material regions and designing them independently, as presented in Ref. [1055], has also served as an inspiration for the development of thermal metamaterials in polygonal configurations [1068]. More recently, a method called extreme anisotropy theory has demonstrated significant potential in designing thermal cloaks of irregular shapes [1074]. Relevant approach has further inspired research into “intelligent thermal metamaterials”, which focus on developing metamaterials that exhibit adaptive responses to environmental changes [1075]. Subsequently, freeform thermal cloaks were realized using topology optimization [1056], which is a fundamental component of numerical optimization theory for thermal cloak design. The detailed information of these optimization theory is introduced as following.

Since the foundational theory of thermal cloaks was established, researchers have increasingly focused on optimizing their designs to overcome the limitations associated with traditional transformation thermotics and scattering-cancellation theory [1076, 1077]. These optimization methods can facilitate the selection of the size and shape of natural bulk materials, thereby aligning them most effectively with the derived thermal properties. Additionally, they can define an optimal device target function for material parameters in situations where scattering-cancellation theory does not yield an analytical solution. Generally, the optimization theory for thermal cloaking can be divided into three aspects: particle-swarm optimization, topology optimization and machine learning. Particle-swarm optimization (PSO) simulates the foraging behavior of birds by iteratively updating each individual’s position in solution space and refining the fitness function, ultimately seeking the optimal solution until the fitness criteria are met. The PSO has been employed for inverse design of diffusion metamaterials in recent years, including applications such as thermal cloaking device [1078] and bilayer thermal sensor [1079]. This approach effectively addresses challenges arising from extreme parameters values by utilizing natural bulk materials. Topology optimization is a versatile algorithm that manages the distribution of components made from natural bulk materials in the design of thermal metamaterials. This method has successfully led to the development of classical thermal cloaks [1080] and multifunctional thermal meta-devices [1081]. The cloak-concentrator device has been further investigated for operation in multiphysical environment, enabling the simultaneous manipulation of thermal and electric fields [1082]. This theory has also introduced a new concept named topological functional cells, which enables the creation of inhomogeneous and anisotropic metamaterials [1056]. This advancement has also significantly broadened the range of anisotropic thermal conductivity [1071]. Machine-learning algorithms offer an alternative approach when design variables are numerous or indirectly related to the performance of metamaterials. Utilizing this algorithm, a four-layer thermal cloak was designed through inverse design, in which an artificial neural network was established to relate thermal conductivities to two objective functions [1083]. Subsequently, a novel design strategy for thermal metamaterials using artificial intelligence was proposed, demonstrating its adaptability to variable environment [1084].

Although most research on thermal cloaking has primarily centered on heat conduction, recent advancements have emerged in cloaking the other two modes of thermal transport. The form invariance of the thermal conduction-convection equation under coordinate transformations was proven in 2015 [1057], leading to the realization of thermal cloaking in this context in 2018 [1085]. Subsequently, through transforming viscosity, the hydrodynamic cloaks were theoretically designed [1086] and experimentally demonstrated [1087]. Thermal radiation, akin to thermal convection, is typically accompanied by heat conduction. The transformation multithermotics theory was introduced in 2020 [1058], which aims to simultaneously control thermal radiation and conduction. It employs the Rosseland diffusion approximation to couple two thermal mechanisms for design of thermal cloak. Besides, based on transformation thermotics, thermal radiative camouflage has been successfully achieved in both 2D [1088] and 3D [1089]. In practice, these three heat transfer modes frequently coexist, leading to the research topic called “Transformation Omnithermotics”, which focuses on manipulating them simultaneously [1090]. In addition to theoretically design of omnithermal cloaking, this study has also contributed to the development of reconfigurable metasurface for thermal illusion [1091].

The primary functions of thermal cloaks can be extended to various related applications, including thermal camouflage [1059], thermal diode [1070], thermal detection using invisible sensor [1067], heat management in electronics, as well as realizations of thermal zero-index cloak [1092] and unidirectional far-infrared cloak [1093]. Thermal camouflage, or illusion, involves generating deceptive signals that build upon the existing capabilities of thermal cloaks, thereby disrupting the observer’s ability to gather accurate reconnaissance. Since its initial investigation in 2014 [1059], research has expanded to explore 3D thermal illusion [1094] and their applicability across multiple physical fields [1095]. Additionally, drawing inspiration from switchable thermal cloaks, a macroscopic thermal diode was developed to facilitate the directional control of heat flux [1070]. Thermal cloaking also plays a crucial role in thermal detection by mitigating interference in the measuring devices during assessments. Its design strategy has been employed to create invisible sensors that obtain more accurate temperature field by minimizing the sensor’s influence [1079]. With the advancement of the microelectronics industry, thermal management within devices has become increasingly critical. Consequently, the relevant principles of thermal cloaking have been applied to guide heat flow in electronic packages [1096, 1097]. Both thermal zero-index cloak and unidirectional far-infrared cloak represent extensions of thermal cloaking, exhibiting significant potential to broaden the range of equivalent thermal conductivity [1092] and offering novel approaches to manipulate infrared radiation [1093]. Fig.45 illustrates a portion of classic experimental cases and recent advancements in thermal cloaking technology.

21.3 Future outlook

While significant advancements have been made in the exploration of thermal cloaks, several challenges remain to be addressed, and additional potential awaits exploration. For instance, the parameters derived from transformation theory often exhibit anisotropic and singular characteristics, indicating the need for continued research to achieve extreme values. Besides, research on cloaking in multiple physical fields is still inadequate and deserves further attention. Although the concepts of thermal and direct current cloaking, as well as invisible sensors have been explored [1059, 1067], there remains considerable potential for further research. By considering the coupling of these phenomena with other physical fields, their practicality for real-world applications can be enhanced. Additionally, several existing designs of thermal cloak exhibit potential for expansion into multi-physical field control, representing another promising direction for future research [1098].

Another significant observation is that most thermal metamaterials focus primarily on macroscopic physics, highlighting a notable gap in research at the microscale and nanoscale [1060, 1099]. In examining diffusion metamaterials at the microscale, the study of phonon diffusion arises as a valuable analytical tool [1100], which warrants further exploration. Additionally, the increased issue of interfacial thermal resistance when considering thermal cloaks at smaller scales [1101] can significantly impact heat transfer [1098]. Given that this phenomenon has often been overlooked in previous research, it is essential to continue investigating it in future studies [1102].

Recently, a theory named diffusive pseudoconformal mapping has established a connection between transformation thermotics and scattering cancellation [1061]. This theory suggests that by constructing a specific angle-preserving mapping, a unified perspective on transformation thermotics can yield parameters identical to those obtained from scattering cancellation in bilayer cloaking design. This advancement effectively addresses the challenges posed by anisotropic parameters resulting from transformation theory, with the potential to fundamentally reshape the design principles of thermal cloaking.

In recent years, advancements across different areas of computer science have revealed significant potential for the design of thermal cloaking when integrated with these innovations. We have discussed the current methodologies for designing thermal cloaks using three optimization theories, highlighting the interdisciplinary nature of this research, which is a crucial direction for future exploration. With the continued application of optimization theories and machine learning, we anticipate that thermal cloak designs will not only develop novel strategies but also expand their practical applications significantly in the future.

21.4 Summary

Thermal cloaking technology represents a significant application of thermal metamaterials, playing essential roles in heat management and military operations. Theoretically, thermal cloaking is achieved through transformation theory or scattering cancellation theory, typically necessitating anisotropic and inhomogeneous parameters. Leveraging effective medium theory (EMT) and other auxiliary frameworks, various design strategies for thermal cloaks have been successfully demonstrated over the past two decades. This review has examined thermal cloaking from six perspectives: (i) early cloaking devices based on classical theories, (ii) key auxiliary theories and multiphysical field cloaking, (iii) advancements in the shapes of thermal cloaks, (iv) theoretical design optimization, (v) exploration of thermal cloaking in three modes of heat transfer, and (vi) related applications. Additionally, we offer a future outlook intended to inspire researchers to advance thermal cloaking technologies, addressing existing challenges and exploring synergies with emerging technologies.

22 Topological thermotics

Qiangkailai Huang^1,2, Ying Li^1,2*

　 ¹State Key Laboratory of Extreme Photonics and Instrumentation, Key Laboratory of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Zhejiang University, Hangzhou 310027, China

　 ²International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China

22.1 Background

The concept of topology in physics originates from the integer quantum Hall effect [1103], where the Hall conductance is equal to integer multiples e^2/h. It was later discovered that this integer part of the Hall conductance is related to a topological invariant of the system, known as the Chern number. This discovery, along with phenomena like the chiral transport of electrons in quantum Hall systems [1104], spark significant interest in topology within quantum systems. Several important theories have since emerged, linking the quantum Hall effect to the breaking of certain symmetries [1105], quantum spin Hall effect [1106, 1107], fractional quantum Hall effect [1108], etc.

The question arises whether this intriguing research area can be extended to classical wave systems. Thanks to the development of classical wave theory, it is possible to derive the Hamiltonian of a system from its governing equations [1109]. This allows for the calculation of band structures in classical wave systems and the analysis of their topological properties. Furthermore, advancements in artificial modulation techniques for classical waves, such as metamaterials [1110], enable researchers to experimentally observe topological phenomena in these systems.

One widely researched property is the bulk-edge correspondence [1111]. In essence, a nonzero topological invariant (nontrivial) in the bulk can predict the emergence of edge states when the system undergoes open boundary conditions (OBCs). These edge states are protected by the nontrivial topology and exhibit robustness, meaning they remain intact despite small perturbations. This mechanism can be utilized to achieve unidirectional transmission [1112], on-chip communication [1113], well-performed laser [1114], etc. These topics have been explored in various classical wave systems, including photonics [1115, 1116], acoustics [1117], circuits [1118], and mechanics [1119].

Unlike other systems, thermal systems inherently possess unique properties such as diffusion and dissipation, making them an interesting platform for studying topology. The combination of thermotics and topology not only brings cutting-edge theory to thermal field regulation but also provides a unique platform for the development of topology. This chapter aims at introducing the development path of topological thermotics, a newly emerging branch of thermal metamaterials, and offer some future outlooks for this area.

22.2 Past to current development

The development timeline of this field follows this sequence: introduction of topological theory into thermotics, experimental verifications in heat conduction, topological thermotics in convection systems, and the investigation of high-order topology [1120]. This article reviews several key works based on this timeline.

Initially, researchers propose incorporating bulk-edge correspondence into diffusion systems. By discretizing Fourier’s law, they derived the Hamiltonian for a diffusion system, enabling the construction of topological tight binding models like the Su−Schrieffer−Heeger (SSH) model [1109]. Subsequent experimental works demonstrated the one-dimensional topological edge state in diffusion systems. One study experimentally verified the bulk-edge correspondence in an SSH model [1121], while another utilized a continuum model to observe topological edge states [1122]. Both of these two works are only based on the platform of purely heat conduction, and the topological edge state can be theoretically explained by the topological invariant Zak phase or winding number. The one-dimensional continuum model and its band structures can be seen in the top right panel of Fig.46. The full band gap appears when the parameter c is varied, and the topological invariant is non-trivial, indicating the presence of topological edge states in OBC.

Beyond heat conduction, topological edge states have also been observed in convection systems. One study developed a one-dimensional spatiotemporal thermal lattice to reveal thermal topological modes, opening a new route for topology in convection systems [1123]. Topology in convective systems also be discovered when we encircle exceptional points (EPs) by modifying two convective parameters [1124]. The bottom right panel in Fig.46 shows this topological convective system and its band structures. Owing to the system’s topological transient property, the system demonstrates good topological robustness, as the isotherms remain unaffected by external perturbations. Furthermore, researchers have extended this kind of topology into a three-dimensional parameter space, discovering Weyl exceptional ring in conduction-convection systems [1125]. Surface-like and bulk topological states can also be observed in this system when the integration surface encloses two coupling topological charges with opposite values.

One-dimensional topology has been well-studied in thermal systems, it is reminiscent to come up with high-order topological state in thermal systems. Researchers explore high-order topological states in various models, including the Kagome lattice [1126–1128], SSH model [1129, 1130], honeycomb lattice [1131], and quadrupole model [1132]. These studies uncover new topological phenomena in high-order topological thermotics, such as corner states, hierarchical band structures, in-bulk edge (corner) states, and more, offering diverse design methods and ideas for further research in this field. We highlight Ref. [1130] as an example for high-order topological thermotics and illustrate it in the bottom left panel of Fig.46. Researchers use nontrivial SSH units to construct topological edge states and corner states, which can be observed in the band structure. The histogram in Fig.46 shows two in-bulk corner states.

In condensed matter physics, non-Hermitian physics has revealed counterintuitive phenomena like break down of the bulk boundary correspondence and non-Hermitian skin effect [1133]. This effect can be predicted by the system’s topological properties in the complex energy plane formed by the real and imaginary parts of the system’s intrinsic energies [1134]. Given that diffusion systems naturally exhibit dissipation, their Hamiltonians are inherently non-Hermitian. Thus, dissipation systems present a promising platform for researching the interplay of topology and non-Hermiticity [1135]. For instance, a one-dimensional non-Hermitian SSH model in a diffusion system is proposed [1136], where the point gap in the complex-energy spectra and the nonzero winding number ensure the occurrence of the non-Hermitian skin effect. As for its application, connecting two edge states in a non-Hermitian thermal system can lead to the construction of a heat funneling device for concentrating thermal energy [1137]. The top left panel of Fig.46 shows the schematic of this heat funneling and its material parameters. The eigenstates distribution and its OBC band structure indicate the occurrence of the non-Hermitian skin effect and the breakdown of bulk-boundary correspondence, respectively. The transient simulation results verify the achievement of heat funneling. In two-dimensional systems, as well as the case in one-dimensional systems, non-Hermitian edge and corner states can also be explained by the occurrence of point gap within the system’s complex-energy spectra [1138].

22.3 Future outlook

The development of topological thermotics is still in its preliminary stages. However, given that dissipation phenomena are very common in nature and the unique characteristics of thermotics, the combination of thermal systems and topology holds significant promise.

One promising direction is the integration of nonreciprocity and topology. Nonreciprocal thermal systems have various potential applications in thermal regulation, such as unidirectional insulated glass, thermal diodes, and energy harvesting. Incorporating topology into nonreciprocal thermal metamaterials can lead to novel application scenarios for topological thermal systems.

Additionally, several aspects of topological non-Hermitian thermal systems remain unexplored. Many topological structures in condensed matter have yet to be realized in thermotics, like three-dimensional topological states. It is hoped that more topological states and non-Hermitian physics will be discovered in thermal fields.

Finally, given that heat generation occurs in many systems, exploring topological states in multi-physical systems can be meaningful. This can be used to achieve thermal protection for electronic components, facilitate solution transport, etc.

In summary, topological thermotics is in its early stages of development. There is substantial room for growth in multiple directions, including structural innovation, discovery of new physical phenomena, practical applications, and integration with other fields, inviting further exploration by researchers.

22.4 Summary

In this chapter, we explore the intersection of topology and thermotics, with a particular focus on topological thermal metamaterials and non-Hermitian topology. One of the core principles discussed is the bulk-edge correspondence, which predicts the emergence of robust topological edge states in systems with non-trivial topological invariants. We first introduce one-dimensional conductive topological systems and then extend the discussion to convective systems. Secondly, we review several works on high-order topological systems. Finally, we introduce non-Hermitian topology, broadening the research area of topological thermotics. Looking ahead, topological thermotics has a long way to go, but we hold an optimistic outlook, suggesting numerous avenues for innovation in structural design, physical phenomena discovery, and practical applications.

23 Non-Hermitian thermal diffusion

Peichao Cao^1,2,3,4, Ying Li^1,2,3,4, Xuefeng Zhu^5,*

　 ¹State Key Laboratory of Extreme Photonics and Instrumentation, Key Lab of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, Zhejiang University, Hangzhou 310027, China

　 ²International Joint Innovation Center, The Electromagnetics Academy at Zhejiang University, Zhejiang University, Haining 314400, China

　 ³Jinhua Institute of Zhejiang University, Zhejiang University, Jinhua 321099, China

　 ⁴Shaoxing Institute of Zhejiang University, Zhejiang University, Shaoxing 312000, China

　 ⁵School of Physics and Innovation Institute, Huazhong University of Science and Technology, Wuhan 430074, China

23.1 Background

In 1928, Gamow [1139] employed a complex eigenvalue framework to elucidate the escape rate of a particle from the nucleus, specifically within the context of alpha decay. In this formulation, the real and imaginary components of eigenvalues correspond to the nuclear resonance energy level and the decay width, respectively. This seminal work is widely regarded as a precursor to the field of non-Hermitian physics. However, since systems that interact with the environment are generally perceived as unstable and unobservable, the non-Hermitian phenomena have garnered limited attentions in the subsequent decades. It was not until 1998 that Bender and Boettcher [1140] elucidated the existence of purely real eigenvalues in non-Hermitian systems that adhere to parity−time (PT) symmetry. This finding challenges the conventional understanding that real eigenvalues can only arise in conservative Hermitian systems.

According to the one-dimensional single-particle Schrödinger equation, a system that exhibits PT symmetry should satisfy the necessary condition V(r)=V^{\ast }(–r) [1141], where V is the complex potential and r the position vector. Unlike the effective Hamiltonian of PT-symmetric systems, which satisfies the commutation relation [H,\mathrm{P}\mathrm{T}]=0, the conjugate symmetry known as anti-PT (APT) symmetry follows the anticommutation relation of \{H,\mathrm{P}\mathrm{T}\}=0 [1142]. Consequently, the eigenvectors of both PT- and APT-symmetric systems are no longer orthogonal, the eigenvalues and eigenmodes simultaneously coalesce at specific points on the Riemann surface, which are known as exceptional points (EPs) [1143].

Turning to the effective Hamiltonian, it is notable that the complex eigenvalues can arise in singular matrices without direct imaginary elements. In this context, non-Hermitian potentials can be interpreted as asymmetric couplings in the physical systems. When the asymmetric coupling is unidirectional, arbitrary-order EPs can be directly attained [1144]. One of the most intriguing outcomes of asymmetric coupling is the non-Hermitian skin effect (NHSE) [1145, 1146], wherein all modes localize at the edges of the system, thereby violating the bulk-boundary correspondence (BBC) relation and significantly modifying the Bloch mode theory applicable to Hermitian cases.

Thermal diffusion, governed by the heat conduction equation derived from Fourier’s law, is characterized by a purely imaginary eigenfrequency when expressed in terms of a wave-like solution, reflecting the diffusive nature of systems. The Hamiltonian governing coupled temperature fields is inherently anti-Hermitian, differing from the Hermitian wave dynamics by a factor of i (i.e., H_{diffusion}=i{H}_{wave}) [1147], with both onsite diffusion rates and offsite coupling strengths represented as imaginary. To replicate the wave-like behavior with momentum in thermal diffusion, real potentials must be introduced, rendering the Hamiltonian of the temperature fields to be non-Hermitian. Like wave dynamics, the non-Hermitian behavior in thermal diffusion can be achieved through two distinct pathways: direct convection [1148] and asymmetric coupling [1149]. This non-Hermitian approach to thermal diffusion presents a novel theoretical framework for the temperature field control, revealing potential applications in thermal phase manipulation and directional heat transfer.

23.2 Past to current development

23.2.1 Convection-driven PT and APT symmetries in thermal diffusion

In wave dynamics, PT/APT symmetry can be readily achieved by introducing non-Hermitian potentials into originally Hermitian systems, such as gain/loss mechanisms in optics [1151], electronics [1152], acoustics [1153] and atomics systems [1154]. Based on this, EP-related single mode lasing [1155], high-precise sensing [1156], wireless power transfer [1157] and unidirectional invisibility [1158] have been extensively explored. In contrast to wave dynamics, the process of constructing non-Hermitian mechanism in heat transfer is reversed: Hermiticity must be introduced into the inherently anti-Hermitian diffusive system to enable momentum-like wave transport.

According to the Fourier’s law for thermal diffusion, convection is associated with the first order derivative term of the temperature field. As a result, the onsite-convection modulation can contribute a real spectrum for the eigenfrequency, corresponding to the thermal phase modulation [as illustrated in Fig.47(a)]. In 2019, the APT symmetric thermal diffusion was firstly discovered in a counter-convection-driven system [1148]. From the coupling equation, the effective Hamiltonian comprises real onsite convection terms and imaginary offsite coupling terms. The EP arises when the convection strength equals to the coupling strength. Below the EP, the system exhibits purely imaginary eigenfrequencies, and the temperature fields dynamically localize with a static phase lag.

However, when convection strength exceeds the EP, APT symmetry is broken. Real eigenfrequency components emerge and the temperature fields evolve continuously with the convection flow. In 2024, the conjugate counterpart of APT symmetry, i.e., PT symmetry, was also realized in thermal diffusion [1150] by the real coupling of temperature fields and the detuning of decay rates. The real coupling between diffusive temperature fields was effectively established via an intermediate strong convection background, while the decay-rate detuning was achieved through the anisotropic design of thermal metamaterials. The system underwent a PT symmetry-breaking phase transition as the convection strength was increased across the EP. Unlike in the APT symmetric case, thermal phase oscillations induced by real potentials were significantly suppressed in both the PT symmetric and PT symmetry-broken regimes.

23.2.2 Asymmetric-coupling-induced non-Hermitian skin effect in thermal diffusion

In mathematics, the effective Hamiltonian of a non-Hermitian dimer system can be expressed as H=[a b;cd ], the EP in such systems occur when the determinant vanishes. In the PT (APT) symmetric systems, the onsite (offsite) elements are imaginary, while the offsite (onsite) elements are real, respectively. The EPs correspond to the scenario when the moduli of onsite and offsite element are equals (i.e., | a|=|b|). Notably, EPs can also arise when both the difference between the onsite elements and the product of offsite elements simultaneously vanish (i.e., ad=0 and bc=0), which implies a unidirectional coupling within the dimer. This asymmetric/unidirectional coupling has been employed to achieve nontrivial energy transport in wave dynamics, particularly in the context of NHSE. To break the spatial symmetry of wave transport and realize such asymmetric coupling, dynamic amplitude modulation is often critical in these setups [1159, 1160].

However, implementing temporal or directional amplitude modulation, as well as unbalanced gain and loss across different pathways, is challenging in thermal diffusion processes. In 2021, an asymmetric coupling toy model was theoretically proposed, based on the inherent non-equivalence between temperature and heat in different materials [1149]. Building on this, the diffusive NHSE was subsequently discovered in the non-Hermitian Su−Schrieffer−Heeger (SSH) model, wherein the eigenstates of all bulk bands localized at the open boundaries. This fundamentally violates the BBC observed in Hermitian systems with periodic boundary conditions, where only edge modes are localized at the boundaries. This unique phenomenon was explained by using the non-Bloch mode theory, in which the Bloch vector is extended into the complex plane [1146, 1161], effectively eliminating the influence of boundary conditions. Additionally, a heat funneling mechanism induced by NHSE was demonstrated, where temperature fields could be concentrated at a target boundary starting from arbitrary initial states. In experimental implementations, the need for precise engineering of material properties was substituted by tailoring the structural parameters for practical realizations [1162]. The observed temperature fields were shown to flow toward the desired boundary [Fig.47(b)]. This sketch was extended to the two dimensional case to explore the higher-order directional thermal regulation recently. The studies showed that the corner-state NHSE exhibited robustness against the Gaussian multiplicative noise [1163].

23.3 Summary and outlook

Thermal diffusion presents a natural platform for investigating the non-Hermitian physics and offers wave-like pathways for thermal management [1164, 1165]. The exploration of PT/APT symmetry, alongside the asymmetric coupling, opens a promising avenue for thermal phase modulation and directional heat transfer. The application of the effective Hamiltonian method has enriched our understanding of temperature field evolution from the perspective of Hermiticity and symmetry. The realization of EPs in various non-Hermitian diffusive systems indicates the potential for high-precision thermal sensing, and the derived thermal topology is anticipated to enable the design of robust functional devices [1166–1168].

Despite these advancements, the precise wave-like control of temperature fields remains to be challenging. Most prior studies have focused on passive systems, where the input temperature field is static, with a zero-frequency response [1147–1150], primarily addressing the transient evolution of temperature fields. Consequently, thermal transport in active scattering systems, particularly those incorporating ports, has been largely overlooked, significantly hindering the application of non-Hermitian thermal diffusion. Therefore, further research is imperative to overcome these challenges and fully harness the potentials of non-Hermitian thermal diffusion, such as the development of scattering matrix method [1169, 1170], the spatiotemporal modulation theory [1171, 1172] and the on-chip integrable technology.

24 Spatiotemporal geometric thermal metamaterials: The principle and application of thermodynamic geometry

Zi Wang, Jie Ren^*

　 Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China

24.1 Background

Thermal metamaterials, with artificially tailored structures, generously provide us a versatile method to control the heat energy and thermal information flows in a way that is previously impossible when using natural materials. In the initial stage, the thermal metamaterial is mainly limited to designing static thermal devices such as thermal diode [1173, 1174], thermal transistors [1175], thermal cloaking [1176, 1177], and camouflaging [1178]. These static thermal metamaterials largely exploits both the nonlinearity of heat conduction and the spatial modulation of system parameters both in the microscopic and macroscopic regime.

However, in recent years, the introduction of external spatiotemporal driving protocols has greatly fertilized this field and enlarged the controllability of thermal flows [1179–1184]. The extra degree of freedom brought by the temporal modulation enables the construction of thermal nonreciprocity [1179], dynamic cooling effect [1182, 1185] and active control over thermal states [1186]. In this chapter, we start by theoretically analyzing the new degrees of freedom offered by driving from a thermodynamic view point. Then we sort out some of the most novel phenomena in spatiotemporal thermal metamaterials, with a focus on the thermal nonreciprocity [1179–1181], geometric heat pump [1183, 1187] and topology [1188]. Finally, we aim to provide some insight and perspective in the future development of this young but interesting field.

Here, we emphasize that, by focusing on the spatiotemporally driven thermal metamaterials, a huge volume of other aspects on thermal metamaterials are largely untouched. For more detailed information, see the recent comprehensive review articles, say, Refs. [1189–1191] on the control of transport by macroscopic thermal metamaterials and Ref. [1192] on topological thermal metamaterials.

24.2 Current status

24.2.1 Theoretical description

We consider the dynamics of diffusion in a bounded range \mathrm{\Omega } of volume | \mathrm{\Omega }|, with the corresponding boundary \mathrm{\partial }\mathrm{\Omega }. By combining Fourier’s law \mathit{J}=-\kappa \mathrm{\nabla } T(\mathit{x},t) and the continuity equation (conservation law) c{\mathrm{\partial }}_{t}T (\mathit{x},t )=-\mathrm{\nabla }\cdot \mathit{J}, the diffusive process is governed by

(70){\mathrm{\partial }}_t T=\mathcal{L}T.

Here, c is the capacity and \kappa is the conductivity. They are assumed to stay positive during the driving protocol. We defined the Laplacian \mathcal{L}:=c^{-1}\mathrm{\nabla }\cdot \kappa \mathrm{\nabla }. We first consider the homogeneous boundary condition (T(\mathrm{\partial }\mathrm{\Omega },t)=0), leaving the time-dependent boundary case to the later discussion. The Laplacian \mathcal{L} is isospectral with the Hermitian operator {\mathcal{L}}_{\mathrm{H}}:=c^{1/2}\mathcal{L}{c}^{-1/2}=c^{-1/2}\mathrm{\nabla }\cdot \kappa \mathrm{\nabla }{c}^{-1/2}. An important property of {\mathcal{L}}_{\mathrm{H}} is that it is negative definite, as can be seen by

(71)

for any function \varphi (\mathit{x}) satisfying the boundary condition \varphi (\mathrm{\partial }\mathrm{\Omega } )=0. By defining the eigenvalue equation for {\mathcal{L}}_{\mathrm{H}} as {\mathcal{L}}_{\mathrm{H}}(\mathit{x})n(\mathit{x})=E_{n}n(\mathit{x}), the corresponding left and right eigenvectors for \mathcal{L} are written as r_n(\mathit{x}):=c^{-1/2}n( \mathit{x}) and l_n(\mathit{x}):=n( \mathit{x})c^{1/2}, with n(\mathit{x}):=n^{\ast }(\mathit{x}).

Apart from the modulation of \kappa and c, miscellaneous dynamic boundary conditions provide addition time-dependent control over thermal transport. The Direchlet boundary condition fixes the boundary T(\mathrm{\partial }\mathrm{\Omega },t) and can be seen as connecting the diffusive system to external thermal reservoirs with controllabel temperature. Another kind of boundary condition is the Neumann condition, which determines the boundary temperature gradient {\mathrm{\partial }}_{n}T(\mathrm{\partial }\mathrm{\Omega },t), with {\mathrm{\partial }}_n being the derivative with respect to the perpendicular direction at the boundary. This can be physically implemented by connecting the system to controllable current sources. In the following, we focus on the Direchlet boundary condition, although other novel boundaries are interesting future topics in spatiotemporal thermal metamaterials.

24.2.2 Geometric heat pump and nonreciprocity

As first shown by Thouless in spatiotemporally driven quantum electron systems, the nontrivial geometric phase manifest itself as a directional current [1196]. However, this geometric effect is not restricted to quantum system. The geometric pump effect was subsequently shown to persist to the classical diffusion regime [1193, 1197, 1198]. The non-vanishing geometric current bequeath the living creatures a way to directionally transport Brownian particles against the violent mesoscopic thermal fluctuations. What is more surprising is that, the geometric pump effect not only regulates the mass transport, but also provides a versatile control over the thermal current itself [1184, 1187, 1199]. Many functional thermal machines [1200] and thermoelectric devices [1194, 1201] can be analyzed under this framework.

Recently, the geometric heat pump effect is generalized to account for the nonreciprocity phenomena [1179–1181] in spatiotemporally thermal metamaterial. As shown in Ref. [1183], the geometric heat pump effect is present in diffusive metamaterial and the non-vanishing geometric heat current is a source for thermal nonreciprocity, even if the thermal transport of the instantaneous steady state is totally reciprocal. Furthermore, the expression for the geometric heat current can be used to classify protocols into no-pumping families and non-trivial ones. Aside from the experiment setup in Ref. [1183], another recent experiment observed this geometric heat pump effect using temporally continuous driving protocols [1202]. We note that the geometric heat pump effect goes much beyond the heat conduction setups. It is also a resource for constructing directed thermal current and thermal nonreciprocity in other physical platforms, like thermal radiation between many-body systems [1203], thermoelectric heat pump [1194] and other multi-physics systems.

The basic theory of the geometric heat pump effect can be cast as the following [1183]. We consider the term {\mathrm{\partial }}_t T and the boundary reservoirs as two sources of perturbations. By defining the inhomogeneous equation for the spatial Green’s function G(\mathit{x},{\mathit{x}}^{\prime } ) as

(72)\mathcal{L}(\mathit{x})G(\mathit{x},{\mathit{x}}^{\prime } )=\delta (\mathit{x}-{\mathit{x}}^{\prime }) ,

with G(\mathit{x},{\mathit{x}}^{\prime }) satisfying the homogeneous boundary condition G(\mathit{x},{\mathit{x}}^{\prime }){|}_{{\mathit{x}}^{\prime }\in \mathrm{\partial }\mathrm{\Omega }}= 0, the general solution to T(\mathit{x},t) is given by

(73)\begin{array}{rl}T(\mathit{x},t)=& {\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{x}}^{\prime }G(\mathit{x},{\mathit{x}}^{\prime }){\mathrm{\partial }}_{t}T({\mathit{x}}^{\prime },t)\\ & -{\int }_{{\mathrm{\partial }}_{\mathrm{\Omega }}}\mathrm{d}{S}^{\prime }T({\mathit{x}}^{\mathbf{\prime }},t){\mathrm{\partial }}_{n^{\prime }}G(\mathit{x},{\mathit{x}}^{\prime }).\end{array}

The first term is the contribution from the bulk \mathrm{\Omega }, while the second term comes from the boundary \mathrm{\partial } \mathrm{\Omega }. If {\mathrm{\partial }}_{t}T({\mathit{x}}^{\mathbf{\prime }},t)=0 (steady state), the first term vanishes and it is obvious that the second term defines the instantanesou steady state temperature profile T_{\mathrm{s}\mathrm{s}}(\mathit{x},t). The Green’s function is actually the inverse of the Laplacian and is related to the eigenstates of \mathcal{L} by

(74)G(\mathit{x},{\mathit{x}}^{\prime } ):={\mathcal{L}}^{-1}={\displaystyle \sum _n{\displaystyle \frac{r_n(\mathit{x})l_n({\mathit{x}}^{\prime })}{E_n}},}

as can be easily seen by observing that it indeed solves Eq. (72). Using the heat current formula \mathit{J}:=-\kappa \mathrm{\nabla }T and Eq. (73), the accumulated heat current density vector splits into two components \mathit{Q}:=\int dt \mathit{J}(\mathit{x},t)={\mathit{Q}}_{\mathrm{d}\mathrm{y}\mathrm{n}}+{\mathit{Q}}_{\mathrm{g}\mathrm{e}\mathrm{o}}. Here, dynamic {\mathit{Q}}_{\mathrm{d}\mathrm{y}\mathrm{n}} is the steady state current {\mathit{Q}}_{\mathrm{d}\mathrm{y}\mathrm{n}}=-\int \mathrm{d}t\kappa \mathrm{\nabla }{T}_{\mathrm{s}\mathrm{s}} and the geometric heat current

(75){\mathit{Q}}_{\mathrm{g}\mathrm{e}\mathrm{o}}(\mathit{x})=- \kappa (\mathit{x}){\displaystyle \int \mathrm{d}t{\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{x}}^{\prime }{\mathrm{\nabla }}_{\mathit{x}}G(\mathit{x},{\mathit{x}}^{\prime }){\mathrm{\partial }}_{t}T({\mathit{x}}^{\prime },t).}}

In the adiabatic limit, we can substitute T({\mathit{x}}^{\prime },t) with its steady state value T_{\mathrm{s}\mathrm{s}}({\mathit{x}}^{\prime },t). If we further parametrize the driving protocol by a time-dependent parameter \mathbf{\Lambda }(t):= [{\mathrm{\Lambda }}_{\mu }( t)], the adiabatic geometric current is given by

(76){\mathit{Q}}_{\mathrm{g}\mathrm{e}\mathrm{o}}^{\mathrm{a}}= {\displaystyle \int \mathrm{d}{\mathrm{\Lambda }}_{\mu }{\mathit{A}}_{\mu }={\displaystyle \propto \mathrm{d}{S}_{\mu \nu }{\mathit{F}}_{\mu \nu },}}

where in the second equality, the Stokes formula is utilized to transform the line integral into the integral of flux across the surface encircled by {\mathrm{\Lambda }}_{\mu }(t) (\mathrm{d}{S}_{\mu \nu } being the surface element). The geometric connection is ({\mathrm{\partial }}_{\mu }:={\mathrm{\partial }}_{{\mathrm{\Lambda }}_{\mu }})

(77){\mathit{A}}_{\mu }:=- \kappa (\mathit{x}){\mathrm{\nabla }}_{\mathit{x}}{\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{d}{\mathit{x}}^{\prime }G(\mathit{x},{\mathit{x}}^{\prime }){\mathrm{\partial }}_{\mu }{T}_{\mathrm{s}\mathrm{s}}({\mathit{x}}^{\prime }),}

and the geometric curvature is given by

(78){\mathit{F}}_{\mu \nu }:={\mathrm{\partial }}_{\mu }{\mathit{A}}_{\nu }-{\mathrm{\partial }}_{\nu }{\mathit{A}}_{\mu }.

The antisymmetric {\mathit{F}}_{\mu \nu } provides a resource to break time-reversal symmetry of transport current. Since the reversal of {\mathit{Q}}_{\mathrm{g}\mathrm{e}\mathrm{o}}^{\mathrm{a}} requires turning around the driving protocol as \mathbf{\Lambda }(t)\to \mathbf{\Lambda }(-t).

The geometric curvature formulation works well in the adiabatic regime, but still leaving the nonadiabatic regime largely unexplored. In a recent work, the finite-time effect on transport is studied, using a geometric metric term to describe the nonadiabatic thermal current [1204]. In contrast to the time-reversal antisymmetric adiabatic geometric current, the nonadiabatic current is time-reversal invariant. It either increase (decrease) the pumped current when it is in the same (opposite) direction to the adiabatic current. In this regime, different coefficients of performance enters the consideration, such as directional current density, thermodynamic efficiency and thermodynamic limits in the spatiotemporal thermal metamaterials. The optimization of heat pump in the finite-time regime is still an interesting question to be systematically addresses in the future studies.

In the above references, two of the three main mechanisms of thermal transport are considered, i.e., conduction and radiation, leaving only the heat convection aside. In fact, by adding a convection term in Eq. (70), the equation of motion becomes totally non-Hermitian and more interesting transport behaviors emerge. In Ref. [1195], the convection together with the diffusion endows the thermal transport an anti-partity-time symmetry, with the symmetry preserving (symmetry broken) phases supporting respectively localized (drifting) behaviors. The convection process introduces an effective Wilis coupling term between the temperature and heat current [1179, 1181, 1205], making the heat transport nonreciprocal. If the thermal distribution is prepared as spatially periodic (wave-like), this Willis coupling enables the diffusive Fizeau drag effect, where the thermal wave propagates at different speed in the opposite directions to the convection flow [1206].

24.2.3 Topology

As thermal systems are dissipative and have fundamentally different physics from closed classical/quantum systems, the construction of topological phenomena in thermal transport is by no way a straightforward task. In pure conduction (diffusive) systems, the thermal wave is dissipative but not coherent. This invalidates the wave topological physics in thermal conduction. Indeed, by directly studying the pure imaginary spectrum of static conduction system, the boundary/corner states in 1D/2D with topologically protected decaying rate are observed [see Ref. [1192]]. These states are localized and non-propagating. To implement topologically protected propagation, the spatiotemporal driving and/or convection need to be incorporated. In Ref. [1195], the anti-PT symmetry breaking ensures a localization-propagation phase transition. In the Anti-PT broken phase, the thermal wavepacket propagates at a constant speed. In Ref. [1188], the spatiotemporal driving and convection-conduction are combined to tailor the topological phases of thermal metamaterials. The topologically protected heat transport is observed therein.

24.3 Future outlook

Above all, we discussed the development status of the spatiotemporal thermal metamaterials. Although many impressive achievements have been made, there are important open questions in this young field. Here, we try to offer a future perspective.

First, the study of the heat pump effect in the finite-time regime is a promising direction in the understanding and construction of thermal metamaterials. The complexity of the real-time dynamical behavior provides more opportunities to optimize the thermal metamaterials with respect to certain tasks, for example, heat pump, dynamic cooling, and thermal diode. Obtaining theoretical constraints on the performance of classes of driving protocol would be an important guidance to the practical design principle. As an example, what is the role played by the symmetry in the spatiotemporal protocols? The miscellaneous powerful machine learning algorithms would also accelerate the study of this direction.

Second, the experimental study of spatiotemporally driven multi-physics setups. The coupling between their several physical components endows controls over one degrees of freedom by manipulating others. An example is Ref. [1185], which uses the controllable electron degrees of freedom to refrigerate its coupled thermal environment.

Third, the huge spectrum of Floquet topological phenomena is to be systematically analyzed and demonstrated in thermal metamaterials. Although in Ref. [1188], the topological edge state is observed in the spatiotemporal thermal metamaterials, the Floquet dynamics is yet largely unexplored. Some examples include the propagating chiral boundary mode, localizations in disordered Floquet systems, and the Anomalous Floquet−Anderson physics.

24.4 Summary

In this chapter, we sketched out the basic theory, recent developments and future outlooks of the spatiotemporal geometric thermal metamaterials. The introduction of spatiotemporal driving enables a versatile construction of geometric heat pump, thermal nonreciprocity, thermal non-Hermitian physics, thermal Fizeau drag, and thermal topological phenomena. These novel behaviors show not only the great power of spatiotemporal driving in controlling thermal transport, but also the huge future potential of this fertile field.

25 Finite-time thermodynamics: A journey beginning with optimizing heat engines

Xiuhua Zhao¹, Yuhan Ma^1,2*

　 ¹School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China

　 ²Graduate School of China Academy of Engineering Physics, Beijing 100193, China

25.1 Background

The utilization of fire and the heat it provides guided early humans through millennia of cultural evolution. Around three centuries ago, the widespread use of heat engines, a key energy technology innovation during the first industrial revolution, rapidly propelled human civilization into the modern era [1207]. Undoubtedly, the development of thermodynamics played a pivotal role in this process. The first and second laws of thermodynamics collectively lead to an interesting result: the efficiency for thermodynamic cycles used to convert heat into work is bounded from above by the Carnot efficiency {\eta }_{\mathrm{C}}\equiv 1- T_c/T_h, where T_h(T_c) is the temperature of the hot (cold) reservoir from which the cycle absorbs (releases) heat [1208, 1209]. {\eta }_{\mathrm{C}} serves as the most fundamental constraint in thermodynamics [1210], determined solely by the temperatures of the reservoirs, independent of the specific properties of the working substance. Classical thermodynamics implies that the reversible thermodynamic cycle required to achieve Carnot efficiency can only be realized in the quasistatic limit. The resulting zero output power makes the Carnot cycle entirely impractical for real-world applications.

Since the industrial revolution, although all practical heat engines satisfy the constraint of Carnot’s theorem; however, there exists a significant gap between their operational efficiencies and {\eta }_{\mathrm{C}} [1211]. This difference arises from the deviation of actual cycles from the quasi-static Carnot cycle, which cannot be adequately described by equilibrium thermodynamics. Therefore, seeking tighter thermodynamic constraints for practical heat engines that reflect the impact of cycle time is a problem of significant practical value, and resolving this requires new theoretical frameworks that go beyond equilibrium thermodynamics. To address this challenge, finite-time thermodynamics emerges with the core objective of bridging the gap between traditional ideal thermodynamic theories and real-world applications, thereby advancing nonequilibrium thermodynamics and providing quantitative results for near-equilibrium processes [1211, 1212].

25.2 Historical development and current trends

25.2.1 Thermodynamic constraint relations

In the 1950s, French physicist Yvon [1213] analyzed the steam cycle in nuclear power plants. Yvon specified the non-quasistatic feature of the cycle as the heat flow caused by the temperature difference between the high-temperature reservoir and the working substance. Using Newton’s law of heat transfer to quantify the heat flow and further identifying cycle power as the optimization objective, he derived the cycle efficiency at maximum power output, now known as efficiency at maximum power (EMP). This new thermodynamic constraint became an important parameter in the later development of finite-time thermodynamics to describe engine performance. Although Yvon’s paper was written in French and initially received little attention, his ideas on engine optimization align with many subsequent works in this field. Around the same period, Chambadal [1214] and Novikov [1215] also investigated similar issues separately, obtaining results consistent with Yvon’s. In 1975, Canadian physicists Curzon and Ahlborn proposed a general endo-reversible Carnot cycle model [1216], simultaneously taking into account the heat flow of the engine in the high-temperature quasi-isothermal process and the low-temperature quasi-isothermal process. They obtained the EMP of the cycle {\eta }_{\mathrm{C}\mathrm{A}}= 1-\sqrt{T_c/T_h}, which was later referred to as Curzon-Ahlborn (CA) efficiency. CA efficiency received considerable attention from experts in nonequilibrium thermodynamics and engineering thermodynamics [1211, 1217], spurring significant development of the emerging field finite-time thermodynamics over the following decades, and leading researchers to systematically study the operation of various practical engines within finite-time thermodynamic cycles [1218–1222].

Beyond the widely used endo-reversible heat engine models in engineering, researchers have explored the EMP of finite-time heat engines from more fundamental perspectives based on various frameworks of nonequilibrium thermodynamics [1223–1232]. For instance, Van den Broeck [1224] applied the linear irreversible thermodynamics to analyze the optimization of steady-state heat engines. Schmiedl and Seifert studied stochastic heat engines with Brownian particles as the working substance, deriving a general expression for EMP [1225]. Tu explored the finite-time operation of the Feynman ratchet, obtaining the EMP of the Feynman ratchet and analyzing the universality of such EMP with respect to Carnot efficiency [1226, 1232]. By introducing the 1 /\tau irreversible entropy generation in finite-time quasi-isothermal processes of duration \tau, Esposito et al. [1228] proved that the EMP of low-dissipation Carnot-like heat engines {\eta }_{\mathrm{L}\mathrm{D}} satisfies {\eta }_{\mathrm{C}}/2\le {\eta }_{\mathrm{L}\mathrm{D}}\le {\eta }_{\mathrm{C}}/\left(2-{\eta }_{\mathrm{C}}\right), where {\eta }_{\mathrm{C}}/2 and {\eta }_{\mathrm{C}}/\left(2-{\eta }_{\mathrm{C}}\right) serve as universal lower and upper bounds of EMP, respectively. In micro scale, Kosloff was the first to investigate the EMP of quantum harmonic oscillator heat engines [1223]. Chen et al. [1231] demonstrated that the extra work of finite-time adiabatic processes (of duration \tau) exhibits 1/ {\tau }^2 scaling, based on which, they found that the EMP of a quantum Otto engine can surpass the upper bound {\eta }_{\mathrm{C}}/\left(2-{\eta }_{\mathrm{C}}\right) of finitetime Carnot engines.

The systematic investigation of EMP has also led to another fundamental issue: the potential constraint relation (also called trade-off relation) between power and efficiency, arising from the irreversibility of the cycle [1233]. Chen and Yan [1234] derived the maximum power for endo-reversible heat engines at a given efficiency. Ryabov and Holubec [1235] collaborated on key advancements in this topic, including the maximum efficiency of steady-state heat engines at arbitrary power within linear irreversible thermodynamics and the approximate power-efficiency constraint relation in the low-power region as well as the near-maximum-power region for low-dissipation heat engines [1236]. In a recent work [1237], one of the authors of this paper and collaborators analytically derived a general constraint relation for power and efficiency across the entire parameter space of low-dissipation heat engines:

(79){\displaystyle \frac{1-\sqrt{1-\tilde{P}}}{2}}\le \tilde{\eta }\le {\displaystyle \frac{1+\sqrt{1-\tilde{P}}}{2-{\eta }_{\mathrm{C}}(1-\sqrt{1-\tilde{P}})}},

where \tilde{P} and \tilde{\eta } are normalized by the maximum power of the heat engine and {\eta }_{\mathrm{C}}, respectively. Some researchers have also derived constraint relations for power and efficiency using thermodynamic geometry [1238] and the general dynamical equations of quantum open systems [1239]. Moreover, recent studies have indicated that when considering the fluctuations in the output power of heat engines, there also exists a constraint relation among power, efficiency and power fluctuations [1240–1242]. In summary, Carnot efficiency, efficiency at maximum power, and the constraint relation between power and efficiency collectively and progressively characterize the performance of finite-time thermodynamic cycles.

25.2.2 Thermodynamic process engineering and optimization

The boundaries of the aforementioned thermodynamic constraint relation indicate the optimal performance of a heat engine at specific fixed parameters, such as the power-efficiency constraint relation, which represents either the maximum efficiency for a given power or the maximum power for a given efficiency. Achieving these boundaries requires that the operation time or control protocol of the thermodynamic processes satisfy certain conditions [1243, 1244]. The concept of thermodynamic process control dates back to the study of thermodynamic geometry [1245–1248]. In the space defined by thermodynamic state variables, thermodynamic length [1245–1247] serves as a metric to quantify the minimum irreversible dissipation that occurs during nonequilibrium processes transitioning between two thermodynamic states [1246, 1247]. Attaining this lower bound requires that the driving of the process adhere to specific criteria, embodying the concept of thermodynamic process control [1249, 1250]. Recently, researchers have extended the near-equilibrium thermodynamic geometry to regions far from equilibrium [1251–1253], enabling broader thermodynamic control.

Physically, the irreversibility of nonequilibrium thermodynamic processes can be quantified by irreversible entropy generation, which, as a process function, depends on the specific driving protocol of the process. Therefore, for a given duration, different driving protocols can lead to varying irreversible entropy generation, resulting in different energy dissipation [1243, 1254, 1255]. Consequently, to minimize dissipation, researchers have developed optimal control strategies for various thermodynamic processes [1243, 1256–1260]. Alternatively, the process time can also be considered as the optimization objective in thermodynamic process optimization, with the goal of achieving the shortest possible duration under certain constraints. For example, when the system interacts with a constant-temperature reservoir, Li et al. [1261, 1262] proposed a strategy named “shortcuts to isothermality” to realize a finite-time isothermal transition by introducing an auxiliary potential. This approach has been experimentally realized [1263] and can be applied to design controllable thermodynamic cycles [1264, 1265]. Furthermore, several studies have also focused on isothermal shortcuts and the control of thermodynamic cycles for quantum systems [1266, 1267].

25.2.3 Unconventional heat engines

After understanding the basic performance constraints and optimal control of various thermodynamic cycles, researchers have started to explore the finite-time performance of unconventional thermodynamic cycles. These studies are primarily motivated by the following two questions: (i) In unconventional scenarios, such as limited total energy supply or presence of additional resources, how should heat engines be optimized? (ii) How does the performance of a heat engine reflect the nonequilibrium thermodynamic properties of its working substance and the coupled reservoirs? For example, conventional heat engines operate with infinite-sized thermal reservoirs at constant temperature. However, when considering reservoirs of finite size, characterized by finite heat capacity, the operations of the heat engine will cause temperature changes in the reservoir [1268]. In this context, the concept of maximum extractable work has been introduced to describe the performance of the heat engine [1269]. Correspondingly, the efficiency at maximum work (EMW) [1268] and the efficiency at maximum average power (EMAP) associated with finitetime cycles [1271] establish the fundamental constraints for heat engines with finite-sized reservoirs. One of the authors of this paper specifically detailed the effects of heat capacity on EMW and EMAP [1272], and further presented a general constraint relation for the power and efficiency of heat engines [1273] with collaborators. It is worth mentioning that the finiteness of the working substance also affects the efficiency of heat engines [1274, 1275], which can be explained by the temperature fluctuations in mesoscopic systems [1275].

Previously, for the sake of theoretical simplicity, noninteracting systems (such as ideal gases, two-level atoms, and quantum harmonic oscillators) were primarily chosen as working substances in heat engines. In the past decade, however, researchers have started exploring whether many-body interacting systems have thermodynamic advantages for constructing heat engines. It was found that interactions can provide collective advantages, allowing irreversible dissipation to increase more slowly than the available power output as the size of the working substance increases, thus improving the efficiency of the heat engine [1276–1280]. Furthermore, many-body interactions may also induce phase transitions in the working substance and several studies have analyzed the performance enhancement of heat engines through phase transitions [1281–1283]. In Ref. [1284], one of the authors of this paper and collaborators proposed a minimal heat engine model with degenerate internal energy levels, which breaks the universal power-efficiency constraint of conventional heat engines, enabling Carnot efficiency at maximum power. Moreover, for micro-scale heat engines, which typically operate in the presence of highly fluctuating energy fluxes, researchers have proposed using fluctuating efficiency to better characterize their performance [1285–1287]. When quantum effects are considered, Scully et al. [1288] revealed that the quantum coherence of the working substance can enhance engine efficiency, prompting further investigations into leveraging quantum coherence as a thermodynamic resource to optimize heat engine performance [1287, 1289–1292].

Recent investigations on active matter have intersected with finite-time thermodynamics, with various researchers proposing the construction of thermodynamic cycles utilizing active matter as working substances or reservoirs [1293–1297]. Pietzonka et al. [1293] proposed a consistent stochastic thermodynamic framework for engines outputting work while being powered by active matter. A most recent work by Wang et al. [1297] explored the optimal control of active matter by extending the traditional thermodynamic geometry framework used for passive systems. Furthermore, discussions on heat engines that violate time-reversal symmetry [1298, 1299], and modifying the interactions between heat engines and thermal reservoirs to regulate the performance of heat engines [1300], also represent a series of beneficial attempts to optimize heat engines. These studies expand the scope of thermodynamic cycle research and provide additional avenues for efficient energy extraction.

25.2.4 Experiments

In contrast to the abundant theoretical advancements, experimental investigations in finite-time thermodynamics are still relatively underdeveloped. In testing the fundamental relations of finite-time thermodynamics, one of the authors of this paper and collaborators utilized an ideal gas platform to measure the irreversible dissipation during the compression process of dry air in contact with a constant-temperature reservoir, rigorously verifying the 1/ \tau scaling of irreversibility in the slow-driving regime [1254]. Based on this platform, Zhai et al. [1301] measured the constraint relation between power and efficiency and the EMP in a complete thermodynamic cycle, thereby validating previous theoretical predictions [1224, 1237, 1243]. Over the past decade, different platforms have successfully implemented various finite-time heat engines and studied their performance. For instance, Blickle and Bechinger [1302] realized a microscopic Stirling engine using colloidal particles in an optical trap. Roßnagel et al. constructed a single-atom heat engine and measured its efficiency [1303]. Martínez et al. [1304] designed a Brownian particle Carnot engine and demonstrated its performance advantages. Krishnamurthy et al. achieved an active Stirling engine operating in a bacterial heat bath [1305]. Besides, some groups have focused on exploring fluctuation relations in micro-scale systems [1306–1311], which are also closely related to finite-time thermodynamics.

25.3 Future outlook

Despite substantial research on finite-time thermodynamic processes and heat engine cycles in the near-equilibrium regime, the regime far from equilibrium still requires further exploration. Currently, most research on heat engine cycles concentrates on the slow-driving regime. Although some studies have begun to investigate fast-driving heat engine cycles [1312, 1313], the inherent need for time-scale separation in theoretical tools suggests that more accurate theoretical approaches must be developed. Characterizing the irreversibility of thermodynamic processes in the fast-driving regime [1314, 1315], as well as optimizing and regulating the performance of heat engines, remain open questions worthy of attention.

Moreover, there has been growing interest in the dynamical processes of information erasure, leading to the identification of the thermodynamic costs associated with finite-time information erasure, known as the finite-time Landauer principle [1316–1320]. This field integrates information thermodynamics [1321] — a novel branch of nonequilibrium thermodynamics — with finite-time thermodynamics. In the future, investigating other thermodynamic constraints in information processing [1322, 1323] and optimizing the thermodynamic costs associated with information management are promising directions, which are of great importance for designing energy-efficient high-performance computing and information-assisted thermodynamic cycles [1289, 1323, 1324].

Applying the framework of finite-time thermodynamics to a variety of practical physical processes and systems is also an intriguing area for future research. For instance, the separation of microscopic particles is essential in fields like biology, medicine, and chemical engineering [1325, 1326]. The ratchet separation scheme proposed by nonequilibrium thermodynamics offers the advantage of not requiring physical entities, enabling high-performance particle separation [1327–1329]. Recently, the authors of this paper along with collaborators studied the energy consumption and optimal control of ratchet separation for Brownian particles [1330]. Applications in battery charging have also emerged, where finite-time thermodynamics helps analyze charging efficiency and charging power [1331]. In addition, the application of finite-time thermodynamic fluctuation relations for efficient free energy estimation remains an active research area [1332–1334], with recent advances in thermodynamic control methods bringing new momentum to the field [1335–1337]. In the nonequilibrium dynamical processes of biological systems, both information processing [1338, 1339] and energy utilization [1340, 1341] present opportunities for further research using finite-time thermodynamics. Many of these theoretical issues require experimental validation, and developing diverse platforms to demonstrate nonequilibrium thermodynamic behaviors is also a crucial direction for future experiments in finite-time thermodynamics.

25.4 Summary

We summarize the historical development of finite-time thermodynamics and review the current state of research over the past two decades in this field, focusing on fundamental constraints of finite-time thermodynamic cycles, optimal control and optimization of thermodynamic processes, the operation of unconventional heat engines, and experimental progress. Exploring and utilizing the constraint relations and optimization methods provided by finite-time thermodynamics across different schemes to enhance energy conversion efficiency and reliability is crucial for the new era of global energy transformation and technological revolution. We conclude this paper with three remarks: (i) It is necessary to develop new theoretical tools for studying fast-driving thermodynamic processes away from equilibrium; (ii) Unconventional heat engines with collective advantages and thermodynamic cycles involving active matter merit further extensive and in-depth research; (iii) Integrating finite-time thermodynamic methodologies and paradigms into a broader range of practical thermodynamic tasks and physical systems, such as micro-particle separation, information processing, and battery performance optimization is a promising and challenging direction for further development in this field.

26 Thermalization in classical lattices: Scaling laws, role of chaos, and real-space analysis

Yue Liu, Yuqi Han, Dahai He^*

　 Department of Physics and Jiujiang Research Institute, Xiamen University, Xiamen 361005, China

26.1 Background

Since the establishment of statistical physics by Maxwell, Boltzmann, and Gibbs, the thermalization of many-body interacting systems has been a fundamental problem of long-standing interest, and to date, this problem is far from being completely solved. The first computer simulation of this problem was conducted in the 1950s by Fermi and his collaborators Pasta, Ulam, and Tsingou [1342]. The model is a one-dimensional chain of N oscillators with nonlinear nearest-neighbor interactions, which is known as the FPUT model. Its Hamiltonian is given by

(80)H={\displaystyle \sum _{i=1}^N{\displaystyle \frac{p_i^2}{2}}+{\displaystyle \frac{1}{2}}(q_{i+1}-q_i{)}^2+{\displaystyle \frac{u}{n}}(q_{i+1}-q_i{)}^n,}

where p_i, q_i and u denote the momentum of ith particle, the position of ith particle and the nonlinear coupling strength, respectively. For n=3, it is referred to the FPUT-\alpha model, while for n=4, it is known as the FPUT-\beta model. By rescaling the nonlinear coefficient and the energy density \epsilon, the distance to the integrable system, namely, the nonintegrable strength can be given by \eta =u{\epsilon }^{(n-2)/2}. The energy of kth mode is defined by E_k=(1 /2 )( P_k^2+{\omega }_k^2{Q}_k^2 ), where {\omega }_k, P_k= \sum _i{p}_i{\varphi }_{ik}, and Q_k= \sum _i{q}_i{\varphi }_{ik} are the frequency, canonical momentum, and canonical position of kth mode, respectively. Here, {\varphi }_{ik} represents the canonical transformation matrix of the system. Due to the nonlinear interactions of the system, it is intuitively expected that the nonlinearly interacting system would eventually reach an energy equipartition state. Contrary to the expectation, numerical results given by Fermi and his collaborators seemingly revealed that the system exhibits quasi-periodic behavior, characterized by energy exchange primarily among a restricted set of modes and a significant dependence on the initial conditions. This phenomenon is known as the FPUT recurrence, and the related problem has puzzled researchers for 70 years [1343–1345]. To delve into the reasons why the FPUT problem deviates from the expected results, statistical physics, nonlinear dynamics, and computational physics have all undergone significant developments [1346]. FPUT-like recurrence phenomena have been observed experimentally across diverse systems, including graphene resonators, nonlinear phonons and photon systems, trapped cold atoms, and Bose−Einstein condensates, underscoring the significance of the thermalization problem [1347].

Basically, previous studies on thermalization primarily focus on three directions. The first one is whether a system can be thermalized, which involves the exploration of ergodicity of the system. The second one is the speed of thermalization, namely, the dependence of thermalization time on the system parameters. The third one involves investigating the underlying microscopic mechanisms of thermalization. These three directions are complementary and inseparable.

The original motivation for the FPUT experiment was verifying the ergodicity and mixing properties of non-integrable many-body systems. Definition of energy in the wave-vector space is employed for numerical verification of equipartition. However, the discovery of the FPUT recurrence phenomenon seemed to suggest that the system fails to reach the equipartition state. Subsequently, people explained this phenomenon based on the soliton scenario [1348], the Chirikov overlap criterion [1349] and the KAM theory [1350]. In Ref. [1351], the authors proposed the concept of metastable states, arguing that the quasi-periodic phenomenon observed in the FPUT numerical experiment is a metastable state. In other words, the system remains in the quasi-periodic state in the FPUT numerical experiment. On longer timescales, the system eventually transitions from the metastable state to a thermalization state. This viewpoint has been verified in recent numerical experiments [1352, 1353]. On the other hand, in recent years, people have attempted to verify the mixing property of the system from the perspective of correlation functions. A research group in Milan numerically observed that correlation functions unexpectedly fail to decay [1354]. Subsequent study by Benettin et al. showed that correlation functions tend to zero over a much longer time [1355]. These results support the validity of the understanding based on metastable states. And it is now generally accepted that even within the quasi-integrable regime, the FPUT model can eventually achieve the thermalization state.

Recently, the primary advancements in classical thermalization lie in the latter two directions, the thermalization speed and the underlying microscopic mechanisms of thermalization. Therefore, in this review, we will revisit the developments in these two directions and introduce the state-of-art progress. Additionally, we will also present recent advances in thermalization indicators.

26.2 Past to current development

26.2.1 Thermalization time

As mentioned above, the second direction discusses the properties of the thermalization time. In the early stages, due to the lack of theoretical tools, numerical simulations were the primary tool to tackle this problem, and distinct results for the behavior of the thermalization time have been proposed. In 1990, Pettini and Landolfi [1356] proposed a stretched exponential law of the form t_{\text{eq}}\sim \exp \left({\epsilon }^{-\gamma }\right) for the thermalization time t_{\text{eq}} and the energy density \epsilon based on numerical simulations and Nekhoroshev’s theorem. In 1995, DeLuca, through numerical experiments [1357], suggested that the equipartition time of the FPUT-\beta model should satisfy the scaling relation t_{\text{eq}}\sim {\epsilon }^{-3} and provided a theoretical explanation in 1999 [1358]. The related theory can be generalized to the case of high-frequency initial excitations [1359], yielding t_{\text{eq}}\sim {\epsilon }^{-2}, which is consistent with numerical results [1361]. However, Berchialla et al. [1360] found in 2004, using the same data as J. DeLuca in 1995, that the equipartition time satisfies a stretched exponential law of the form t_{\text{eq}}\sim \exp \left({\epsilon }^{-1/4}\right). In 2011 and 2013, Benettin et al. [1352, 1355] found through numerical experiments that in the FPUT-\beta model, the thermalization time undergoes a transition from a stretched exponential law t_{\text{eq}}\sim \exp \left({\epsilon }^{-1/4}\right) to a power law t_{\text{eq}}\sim {\epsilon }^{-9/4}, while in the FPUT-\alpha \beta model, the thermalization time undergoes a transition from a stretched exponential law t_{\text{eq}}\sim \exp \left({\epsilon }^{-1/8}\right) to a power law t_{\text{eq}}\sim {\epsilon }^{-9/4}. Up to this point, there is a lack of rigorous theory for the qualitative laws governing thermalization time, and there is no consensus on which law the thermalization time should follow.

In 2015, Onorato et al. [1362] applied the wave-turbulence (WT) theory to investigate the behavior of thermalization time. According to this theory, they found that the thermalization time of the small-size FPUT-\alpha model scales as t_{\text{eq}}\propto {\alpha }^{-8} in the quasi-integrable region. In 2018 [1363], they observed that for small systems in the quasi-integrable region, the thermalization time of the FPUT-\beta model scales as t_{\text{eq}}\propto {\beta }^{-4}, while in the large-N limit, it scales as t_{\text{eq}}\propto {\beta }^{-2}. Furthermore, it has been found that in the quasi-integrable region, the thermalization time scales as t_{\text{eq}}\propto {\eta }^{-2} in the thermodynamic limit based on the WT theory [1364–1369]. These results provide important insights into the properties of thermalization [1370]. Additionally, the introduction of the WT theory allows for a more theoretical approach to studying thermalization, enabling the verification of numerical results through analytical calculations.

Within the framework of the WT theory [1370], thermalization is achieved through irreversible energy transfer mediated by non-trivial resonant wave-wave interactions in the spectrum. By introducing complex coordinates a_k=(P_k+i{\omega }_k{Q}_k)/\sqrt{2{\omega }_k}, the nth-order nonlinear interaction can be expressed as of the q-wave interactions. The main aim of the WT theory is to find the the kinetic equation of the power spectrum density {\mathrm{\Phi }}_k=⟨a_k{a}_k^{\ast }⟩, with the assumption that a_k is independent with each other and satisfies Gaussian statistics. Therefore, the high-order correlation can be rewritten as a function of low-order correlation completely. The resulting kinetic equation for {\mathrm{\Phi }}_k is given by

(81){\dot{\mathrm{\Phi }}}_k={\eta }^2\mathcal{F}(\mathit{\omega },\mathbf{\Phi }),

where \mathcal{F} contains the information of wave-wave interactions and multi-wave resonance conditions. Eq. (81) implies that thermalization time is proportional to {\eta }^{-2}, namely,

(82)t_{\text{eq}}\propto {\eta }^{-2}=u^{-2}{\epsilon }^{-(n-2)}.

Note that thermalization occurs only when all the modes can be connected through non-trivial resonant wave-wave interactions. For any n\ge 4, Eq. (82) is valid in the thermodynamic limit. However, in the case of n=3, the integrability of Toda lattices precludes non-trivial three-wave resonances, resulting in the dominance of higher-order interactions in thermalization processes.

26.2.2 Microscopic mechanisms of thermalization

A fundamental question in classical statistical mechanics concerns the precise relationship between thermalization and chaos, which has attracted significant attention in the past few decades [1356, 1371–1375]. In 1990, Pettini et al. defined an equipartition indicator [1356, 1372] and observed distinct dynamical behaviors for this indicator across two distinct timescales. Thus, the relaxation time is determined in terms of the transition point. They found that the relaxation time exhibits two distinct behaviors at different energy scales. Meanwhile, they found that the largest Lyapunov exponent exhibits similar behaviors at closely the same transition point. This reveals that the chaotic property of the system is relevant with its relaxation properties. Similarly, in 2018, the transition point of the double scaling behavior obtained from the WT theory coincides with the transition point from weak to strong chaotic regime obtained by the Chirikov criterion. This indicates that the microscopic orbital instability of the system can significantly affect the relaxation properties of the system [1363]. In 2019, Danieli et al. [1373, 1374] defined the ergodization time of the rotor systems and Klein−Gordon lattices, and compared the ergodization time with the Lyapunov time (the inverse of the largest Lyapunov exponent). They found that the ergodization time and the Lyapunov time with a surprisingly large ratio (greater than {10}^8), exhibit different monotonic behaviors. In 2020, Ganapa et al. [1375] defined an equilibration time to quantitatively describe the relaxation process of the system. Numerical results demonstrate that while the scaling laws governing the equilibration time and the Lyapunov time differ within a specific energy density range, they exhibit a remarkable degree of proximity. The authors believe that there is a strong correlation between the relaxation properties and chaotic properties of the system. However, by this definition, the Toda model shows the tendency to equilibrium, which contradicts the expectation and requires further discussions. Specifically, the precise quantitative relationship between the characteristic timescales governing relaxation and chaos remains elusive. Similar questions have also attracted much attention in the quantum thermalization [1376–1380], because it involves the relationship between microscopic properties (chaos) and macroscopic properties (relaxation).

Based on the geometrization of dynamics and the self-consistent phonon theory, an analytical approach has been developed to derive the Lyapunov time [1381], which shows the Lyapunov time exhibits the scaling behavior t_{\text{L}}\propto {\eta }^{-2} in the quasi-integrable region and t_{\text{L}}\propto {\eta }^{-1/n} in the strongly nonintegrable region. In the quasi-integrable region, the scaling exponent coincides with the behavior of the thermalization time given by Eq. (82), which means that the thermalization time is proportional to the Lyapunov time

(83)t_{\text{eq}}=\alpha t_{\text{L}},

where the coefficient \alpha depends on the initial conditions and the way to define the thermalization time (see Section 26.2.3).

Furthermore, in the regime beyond the quasi-integrable, the thermalization time can be calculated by the real-space method (see Section 26.2.3). As one can see in Fig.50, the thermalization time is proportional to the Lyapunov time not only in the quasi-integrable regime but also in the regime beyond the quasi-integrability.

Eq. (83) bridges the microscopic chaotic dynamics and the macroscopic thermalization process, providing a new perspective on the relationship between chaos and thermalization. However, for the model of the asymmetric interaction such as the FPUT-\alpha model and the FPUT-\alpha \beta model, the situation becomes more complicated, where more factors such as the boundary condition should be taken into consideration. And the proportional relationship could be an approximation of a more rigorous theory.

26.2.3 Thermalization indicators

In 1985, to investigate the thermalization time, Livi et al. [1382] proposed the spectral entropy as the thermalization indicator based on the energy distribution of the system

(84)s(t)=-{\displaystyle \sum _{k=1}^N{\rho }_k(t)\ln {\rho }_k(t),{\rho }_k(t)={\displaystyle \frac{{\overline{E}}_k}{{{\displaystyle \sum }}_{k=1}^N{\overline{E}}_k}},}

where {\overline{E}}_k represents the time-average energy of the kth mode. When the system reaches the equipartition state, the spectral entropy reaches its maximum value s_{max}=\ln N. In subsequent studies [1352, 1356], the spectral entropy is normalized to the range [0,1 ]

(85)h(t)={\displaystyle \frac{e^{s(t)}}{N}},\text{and}h( t)={\displaystyle \frac{s_{max}-s(t)}{s_{max}-s(0)}},

which refer to the “effective number of degrees of freedom” and the normalized spectral entropy, respectively. In order to characterize the contribution of high-frequency modes, Benettin et al. [1352] changed the summation range to [N/2,N] and modified the normalization factor accordingly. With that the spectral entropy and its derivatives have been widely used in studies of thermalization in classical lattices [1346].

However, the spectral entropy, being inherently defined in terms of mode energy in the wave-vector space, becomes ill-defined beyond the quasi-integrable regime. This motivates the exploration of alternative approaches, such as analyzing thermalization directly in real space. Several approaches have been proposed to describe thermalization behavior in real space. Parisi [1384] utilized the nearest-neighbor correlation in real space to explore the relaxation time through its logarithmic time derivative. This method reveals that the thermalization time exhibits an stretched exponential behavior as a function of anharmonicity. Meanwhile, it has been observed that the ergodization time defined by analyzing the fluctuations of observables shows qualitatively distinct behavior with the Lyapunov time [1373, 1374]. Additionally, the study on the thermalization of local observables [1375] indicates that the scaling behavior of equilibration time is contingent on initial conditions, highlighting the absence of universality.

Recently, in terms of the indicator proposed by Parisi

(86)\mathrm{\Delta }={\displaystyle \frac{⟨{\displaystyle \sum _n{p}_n{p}_{n+1}}⟩}{⟨{\displaystyle \sum _n{p}_n^2}⟩}},

a scheme has been developed to study the thermalization of the classical Hamiltonian chain of interacting oscillators in real space [1383]. This indicator goes to zero in the thermal state. For the harmonic lattice, the generalized Gibbs ensemble dictates the values of \mathrm{\Delta }, which are generally nonzero. By quenching the harmonic system to a nonintegrable model, the nonequilibrium initial conditions can be prepared and one can determine the thermalization time by analyzing how \mathrm{\Delta } approach to zero. This approach, independent of the wave-vector space, is crucial for exploring regimes beyond quasi-integrability, where mode energy definitions can become ambiguous. This method can be applied to the FPUT-\beta model as an example. As shown in Fig.50, the thermalization time shows a double scaling behavior with respect to the nonintegrable strength \eta. In the quasi-integrable region, the thermalization time scales as t_{\text{eq}}\propto {\eta }^{-2}, which is consistent with the WT theory. In the strongly integrable region, the thermalization time scales as t_{\text{eq}}\propto {\eta }^{-1/4}, which is the same as the Lyapunov time. Moreover, one can find that the thermalization time defined in the real space is again proportional to the Lyapunov time.

Note that the scaling behavior of the thermalization time is more concerned than seeking for its exact numeric value, which can only be obtained for h(t_{\text{eq}} )=1. In previous studies investigating thermalization in the wave-vector space (e.g., see Ref. [1352]), the thermalization time is defined as the time when the “effective number of degrees of freedom’’ h reaches 0.5. The reason is when the scaling behavior is concerned different thresholds for h (chosen to define thermalization time) yield the same scaling properties. The thermalization time that defined in real space serves as a characteristic time, which does not mean that the thermal state is reached at this time, nor is it equal to the thermalization time obtained in the wave-vector space. In this sense, the absolute value of the thermalization time is not relevant. What matters is that the (scaling) behavior of the thermalization time in real space is consistent with that in the wave-vector space, e.g., the -2 scaling law in the quasi-integrable region. This represents an important aspect in comprehending the mechanism of thermalization.

26.3 Future outlook

First, it is crucial to consider the impact of intrinsic excitations on thermalization. Up to now, the majority of studies on thermalization has been based on models of the nearest-neighbor interaction. When the potential term is changed, e.g., adding on-site potentials or altering long-range interactions, relevant intrinsic excitations are possibly generated. For instance, in studies of heat conduction, the emergence of intrinsic localized modes, such as discrete breathers, can significantly affect the transport properties of the system. Discrete breathers interact with phonons or solitons, suppressing heat conduction through the system. In a similar vein, intrinsic excitations should play a role in thermalization processes. Further investigations are necessary to fully understand the impact of intrinsic excitations.

Second, the selection of observables is significant in studies of thermalization, particularly with regard to the choice of the Toda model as a reference. In previous studies, researchers have typically used the energy of normal modes as an observable, focusing on the mixing behavior of phase trajectories in non-integrable systems on the harmonic tori. Similarly, the Toda model, as a unique classical nonlinear integrable model, can also serve as a reference, with its integrals of motion derived through the Lax method. This approach offers a distinct advantage when analyzing asymmetric models, such as the FPUT-\alpha \beta model. However, when adopting the Toda model as a reference, it is important to consider whether certain characteristic times, such as thermalization time, retain the same time scale as in previous results.

26.4 Summary

This chapter focuses on recent advancements in understanding of thermalization in classical lattices of interacting oscillators. While the existence of a thermalization state has been extensively studied, this review delves more into the following aspects: the characteristic timescale of thermalization, the underlying microscopic mechanisms, and the development of suitable thermalization indicators.

The study of thermalization time has evolved from early numerical experiments, with various scaling laws proposed. With the application of the WT theory, a universal inverse-square relation between T_{eq} and \eta has been proposed. The microscopic mechanisms of thermalization is deeply relevant with the chaotic nature of the system. In the quasi-integrable region, the Lyapunov time follows t_L\propto {\eta }^{-2}, while in the strongly non-integrable region, it scales as t_L\propto {\eta }^{-1/n}. This suggests that the thermalization time is proportional to the Lyapunov time, with the proportionality constant \alpha depending on the initial conditions and the method used to define the thermalization time. This review further explores the development of thermalization indicators, with a specific emphasis on a real-space analysis. By taking the FPUT-\beta model as an example, the aforementioned scaling behavior and proportional relationship have been successfully reproduced in real space using this method.

A promising avenue for future study lies in exploring the influence of intrinsic excitations, such as discrete breathers, on thermalization processes. Furthermore, one can consider the selection of observables, as they can significantly impact the observed scaling behavior of thermalization time.

27 Closing remarks

Yuguang Qiu, Jiping Huang^*

　 Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China

The field of thermal metamaterials has evolved into a multidisciplinary frontier bridging thermodynamics, condensed matter physics, quantum engineering, and advanced manufacturing. By establishing universal design principles across scales – from the nanoscale (Chapters 2−4) to macroscopic transformation thermotics (Chapters 6−8) – researchers have achieved unprecedented control over heat flow. Breakthroughs such as silicon nanomeshes suppressing thermal conductivity below the amorphous limit [Chapter 2.2.5, Fig.2(c)] and folded-space thermal lenses demonstrating negative thermal conductivity (Chapter 6.2.2) now challenge classical Fourier paradigms. The emergence of non-Fourier regimes (Chapter 9) and fractional-order models (Chapter 11) has further enabled precise descriptions of ultrafast thermal relaxation in 2D heterostructures, validated by AI. At the quantum frontier, topological thermal effects including chiral edge states (Chapter 22.3) and non-Hermitian exceptional points (Chapter 23.4) have transitioned from theoretical curiosities to laboratory realities, achieving high directional rectification and substantial enhancements in thermal sensitivity at room temperature.

Despite these advances, critical challenges persist at the intersection of theory and experimentation. The wave-particle duality of phonons remains inadequately modeled in sub-10 nm structures, where coherent transport mechanisms (Chapter 3.4) compete with Anderson localization under strong quantum fluctuations (Chapter 11.2). Dynamic reconfigurability – a cornerstone for practical applications – still suffers from limited tunability (Chapter 7.3) and poor cycling endurance (phase-change materials, Chapter 8.2). Manufacturing bottlenecks are particularly acute for atomic-scale thermal cloaks (Chapter 4.2.2), where the feature nano-sizes required for Al₂O₃ rings [Fig.10(a)] starkly contrast with current nanofabrication capabilities (Chapter 18).

Looking ahead, the convergence of topology, quantum phenomena, and machine learning heralds transformative opportunities. Integrating topological thermotics (Chapter 22) with superconducting qubit architectures (Chapter 3) could pioneer entanglement-based thermal memories, while inverse-design algorithms (Chapters 19−20) may enable neuromorphic thermal circuits that emulate biological homeostasis. Beyond terrestrial applications, radiation-engineered metasurfaces (Chapter 15.3) coupled with thermionic converters (Chapter 17.4) offer viable pathways for Mars exploration systems to harness diurnal thermal cycles. Ultrafast spectroscopy advancements are poised to unlock attosecond-resolution studies of phonon squeezing in twisted graphene (Chapter 16.2.3), potentially rewriting quantum thermal transport rules. As these threads converge, thermal metamaterials will transcend their traditional role as passive heat managers, emerging instead as active participants in the quantum information revolution and humanity’s expansion into deep space.

In general, this Roadmap presents the development of the field of thermal metamaterials through Chapters 2−26, each offering a unique perspective. Each chapter discusses the past evolution, current frontiers, and future outlook of a specific branch. It not only reflects the development of smaller sub-branches, but also brings together a macro picture of their interweaving as a whole. In terms of content, these chapters can be divided into multiple heat transfer modes, such as heat conduction, heat convection, heat radiation, and multi-mechanism coupling. They span a broad range of topics, from single thermal systems to multi-physics interactions involving thermoelectrics, optoelectronics, and more. Some chapters delve into specific properties of physical fields, such as nonlinearity, topology, and non-Hermitian concepts, guiding the reader through the depth and breadth of thermal metamaterials science. Others focus on the fundamental thermodynamic principles, laying a solid foundation for the future development of thermal metamaterials.

This review not only provides a comprehensive overview of the past and present of the thermal metamaterials field, but also offers deep insight and guidance for its future directions. While we have made every effort to be thorough in our review, it is important to acknowledge that some aspects of thermal metamaterials research may not have been covered. Therefore, we look forward to future studies that, while building upon the existing work, will both deepen exploration in established areas and boldly venture into new, unexplored directions, contributing to the continued flourishing and advancement of the field of thermal metamaterials.

Anticipating the future, the development of thermal metamaterials lies not only in further theoretical advancements but also in fostering international collaborations that transcend geographical and disciplinary boundaries. As we continue to push the boundaries of what is possible in thermal management, it is crucial that researchers from different corners of the world come together to share knowledge, resources, and expertise. Interdisciplinary cooperation between academia, industry, and government bodies will be key to solving the complex challenges that lie ahead, from the practical implementation of advanced thermal designs to addressing global sustainability issues. By working together across nations and sectors, we can accelerate innovation, exchange novel ideas, and bring thermal metamaterials from laboratory breakthroughs to real-world applications. This Roadmap functions both as a mirror reflecting our current standing and as a clarion call for the international community to unite in shaping the future trajectory of this transformative domain, thereby ensuring a brighter future for all of humanity.

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