In the existing literature of thermal metamaterials or metadevices, many properties or functions are designed via coordinate transformation theory (transformation thermotics), including thermal concentrating and cloaking. But other properties or functions, say, thermal transparency, are designed by using theories differing from the transformation thermotics. Here, we put forward an effective medium theory in thermotics by considering anisotropic layered/graded structures, and we reveal that the theory can unify transparency, concentrating, and cloaking into the same theoretical framework. Furthermore, the theory not only gives the criterion for transparency, concentrating, and cloaking, but also helps to predict a type of ellipses-embedded structures which can achieve transparency, concentrating, and cloaking, respectively. The prediction is confirmed by our finiteelement simulations and/or experiments. This work provides a different theory to understand and design thermal metamaterials or metadevices, which might be extended to other disciplines, such as optics/electromagnetics and acoustics.

In this paper, we review some recent achievements in thermal metamaterials, including novel thermal devices, simplified experimental method, macroscopic thermal diode based on temperature-dependent transformation thermotics, and the important role that soft matters play in the experimental confirmations of thermal metamaterials. These works pave the developments in transformation mapping theory and can surely inspire more designs of thermal metamaterials. What is more, some approaches provide more flexibility in controlling heat flow, and they may also be useful in other fields that are closely related to temperature gradient, such as the Seebeck effect and many other domains where transformation theory is valid.

Manipulating thermal conductivities are fundamentally important for controlling the conduction of heat at will. Thermal cloaks and concentrators, which have been extensively studied recently, are actually graded materials designed according to coordinate transformation approaches, and their effective thermal conductivity is equal to that of the host medium outside the cloak or concentrator. Here we attempt to investigate a more general problem: what is the effective thermal conductivity of graded materials? In particular, we perform a first-principles approach to the analytic exact results of effective thermal conductivities of materials possessing either power-law or linear gradation profiles. On the other hand, by solving Laplace's equation, we derive a differential equation for calculating the effective thermal conductivity of a material whose thermal conductivity varies along the radius with arbitrary gradation profiles. The two methods agree with each other for both external and internal heat sources, as confirmed by simulation and experiment. This work provides different methods for designing new thermal metamaterials (including thermal cloaks and concentrators), in order to control or manipulate the transfer of heat.