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Formal analysis, Writing - orginal draft [1]

[1]Corresponding author.

1]organization=Department of Physics, State Key Laboratory of Surface Physics,
and Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan
University, city=Shanghai, postcode=200438, country=China

1]organization=Graduate School of China Academy of Engineering Physics,
city=Beijing, postcode=100193, country=China

# Radiative metamaterials based on effective-medium theory

Haohan Tan<EMAIL_ADDRESS>[  Liujun Xu  [

###### Abstract

Thermal metamaterials have made significant advancements in the past few
decades. However, the concept of thermal metamaterials is primarily rooted in
the thermal conduction mechanism, which has consequently restricted their
application scope. It is imperative to consider thermal radiation, another
crucial thermal transport mechanism, particularly in high-temperature regimes,
when designing thermal devices. In this review paper, we present the
advancements in this area, with a specific focus on research conducted using
the effective-medium theory. Additionally, we explore the potential
applications of radiative thermal metamaterials and discuss prospective
research directions from a microscopic perspective for future investigations.

###### keywords:

Radiation

##  1 Background

The research on diffusion metamaterials dates back to 2008 [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12]. In that year, Huang et al. proposed thermotics. It is
worth noting that prior to the proposition of thermotics, Huang et al.’s
research focus was not on metamaterials but on soft matter [13, 14, 15, 16],
such as colloidal particles [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29], colloidal crystal [30, 31, 32, 33, 34, 35, 36], and colloidal ferrofluids
[37, 38, 39, 40, 41]), electrorheological fluid [42, 43, 44, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54], and electrorheological solid [55, 56, 57],
magnetorheological fluid [58, 59, 60, 61]. They also ventured into the
research of water, including the movement of water molecules [62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73] and phase transition [74, 75, 76],
electromagnetism, including electrics [77, 78, 79, 80, 81, 82, 83, 84, 85, 86,
87, 88], nonlinear response [89, 90, 91, 92, 93, 94, 95, 96], dispersion [97,
98, 99], electrorotation [100, 101, 102], transformation electrics [103, 104],
ferrofluids [105, 106, 107, 108, 109, 110, 111], and other fields.
Furthermore, they were also interested in sociology [112, 113, 114, 115, 116,
117, 118, 119, 120, 121, 122, 123, 124, 125], econophysics, including the
stock market [126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137], and
other aspects [138, 139, 140, 141, 142, 143], statistics [144, 145], condensed
matter physics [146], and energy engineering [147, 148]. Perhaps it is the
study of optics, including optical materials [149, 150, 151, 152, 153, 154,
155, 156, 157, 158] and nonlinear response [159, 160, 161, 162, 163, 164, 165,
166, 167, 168, 169], and acoustics [170, 171, 172, 173] that paved the way for
the eventual birth of thermotics.

Since then, the research in the field of diffusion metamaterials has made
significant progress [174, 175, 176, 177, 178, 179, 180, 181, 182].
Researchers have designed numerous devices with novel functions, such as
thermal cloaks [183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194,
195], concentrators [196, 197], rotators [198], sensors [199, 200, 201, 202,
203], illusions [204, 205, 206, 207, 208, 209], transparency [210, 211, 212,
213, 214, 215], and expanders [216, 217]. The development of metamaterials in
the thermal field began with research on thermal conduction [218, 219, 220,
221, 222, 223, 224, 225, 226, 227, 228]. However, thermal radiation [229, 230,
231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241], which is an important
mode of thermal transport alongside conduction and convection [242, 243, 244],
cannot be ignored in many cases. Finding ways to realize similar functions
while considering this mechanism is an unavoidable problem. However, the
nonlinearity of Stefan’s law has posed significant obstacles to research in
this field. Furthermore, it is unknown whether the dynamic equations for
thermal conduction and radiation satisfy transformation invariance. Therefore,
the design of thermal metamaterials considering conduction and radiation
remains an unexplored topic, even many years after the proposition of thermal
metamaterials. Xu et al. were the first to design diffusion metamaterials for
manipulating thermal radiation. For convenience, they did not use Stefan’s law
but instead employed the Rosseland diffusion approximation. Based on this
method, they designed devices with novel functions such as cloaks,
transparency, and expanders.

Figure 1: Schematic diagrams of (a) thermal transparency, (b) thermal cloak,
and (c) thermal expander. (d) and (e) qualitatively show the radiative
emittance $j$, conductive flux $J_{con}$, and radiative flux $J_{rad}$ as a
function of temperature $T$. Adapted from Ref. [240]

##  2 Effective-medium theory under Rosseland approximation

Xu et al. investigated a passive and steady process of heat transfer, focusing
on the total heat flux $\bm{J}_{\text{total}}$, which comprises the conductive
flux $\bm{J}_{\text{con}}$ and the radiative flux $\bm{J}_{\text{rad}}$. This
heat flux satisfies the divergence-free condition:

$\bm{\nabla}\cdot\bm{J}_{\text{total}}=\bm{\nabla}\cdot\left(\bm{J}_{\text{con}}+\bm{J}_{\text{rad}}\right)=0.$
(1)

The conductive flux $\bm{J}_{\text{con}}$ is given by:

$\bm{J}_{\text{con}}=-\kappa\bm{\nabla}T,$ (2)

where $\kappa$ represents the thermal conductivity. On the other hand, based
on the Rosseland diffusion approximation, the radiative flux
$\bm{J}_{\text{rad}}$ is expressed as:

$\bm{J}_{\text{rad}}=-\gamma T^{3}\bm{\nabla}T,$ (3)

Here, $\gamma$ (given by $\gamma=16\beta^{-1}n^{2}\sigma/3$) can be considered
as the radiative coefficient. In this expression, $\beta$ corresponds to the
Rosseland mean extinction coefficient, $n$ represents the relative refractive
index, and $\sigma$ is the Stefan-Boltzmann constant ($\sigma=5.67\times
10^{-8}\,{\rm Wm}^{-2}{\rm K}^{-4}$).

Xu et al. further consider a three-dimensional core-shell structure (Fig.
1(a)) which consists of a core with thermal conductivity $\kappa_{c}$,
Rosseland mean extinction coefficient $\beta_{c}$, and relative refractive
index $n_{c}$ (radiative coefficient $\gamma_{c}$), coated by a shell with
corresponding parameters $\kappa_{s}$, $\beta_{s}$, and $n_{s}$ (radiative
coefficient $\gamma_{s}$). The subscript $c$ (or $s$) denotes the core (or
shell). The semi-axis lengths of the core and shell are $\lambda_{ci}$ and
$\lambda_{si}$, respectively, where $i=1,\,2,\,3$. Assuming that the ratio
$\gamma/\kappa$ of the core-shell structure is a constant $\alpha$,
specifically $\gamma_{c}/\kappa_{c}=\gamma_{s}/\kappa_{s}=\alpha$, Equation
(1) can be rewritten as

$\bm{\nabla}\cdot\left(-\kappa\bm{\nabla}T-\alpha\kappa
T^{3}\bm{\nabla}T\right)=\bm{\nabla}\cdot\left(-\kappa\left(1+\alpha
T^{3}\right)\bm{\nabla}T\right)=\bm{\nabla}\cdot\left(-\kappa\bm{\nabla}\left(T+\alpha
T^{4}/4\right)\right)=0.$ (4)

By performing a variable substitution $\varphi=T+\alpha T^{4}/4$, we obtain

$\bm{\nabla}\cdot\left(-\kappa\bm{\nabla}\varphi\right)=0.$ (5)

Therefore, the strongly nonlinear equation (Eq. (1)) can be transformed into a
linear equation (Eq. (5)).

To proceed, Xu et al. introduced ellipsoidal coordinates
$\left(\rho,\,\xi,\,\eta\right)$, which are defined by the following
equations:

$\left\\{\begin{array}[]{l}\frac{x^{2}}{\rho+\lambda_{1}^{2}}+\frac{y^{2}}{\rho+\lambda_{2}^{2}}+\frac{z^{2}}{\rho+\lambda_{3}^{2}}=1\,\left({\rm
confocal\ ellipsoids}\right)\\\
\frac{x^{2}}{\xi+\lambda_{1}^{2}}+\frac{y^{2}}{\xi+\lambda_{2}^{2}}+\frac{z^{2}}{\xi+\lambda_{3}^{2}}=1\,\left({\rm
hyperboloids\ of\ one\ sheet}\right)\\\
\frac{x^{2}}{\eta+\lambda_{1}^{2}}+\frac{y^{2}}{\eta+\lambda_{2}^{2}}+\frac{z^{2}}{\eta+\lambda_{3}^{2}}=1\,\left({\rm
hyperboloids\ of\ two\ sheets}\right)\end{array}\right.$ (6)

In these equations, $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are three
constants that satisfy
$\rho>-\lambda_{1}^{2}>\xi>-\lambda_{2}^{2}>\eta>-\lambda_{3}^{2}$. Equation
(5) can be expressed in ellipsoidal coordinates as:

$\frac{\partial}{\partial\rho}\left(g\left(\rho\right)\frac{\partial\varphi}{\partial\rho}\right)+\frac{g\left(\rho\right)}{\rho+\lambda_{i}^{2}}\frac{\partial\varphi}{\partial\rho}=0,$
(7)

where
$g\left(\rho\right)=\sqrt{\left(\rho+\lambda_{1}^{2}\right)\left(\rho+\lambda_{2}^{2}\right)\left(\rho+\lambda_{3}^{2}\right)}$.
Since Eq. (7) has the solution:

$\varphi=\left(u+v\int_{0}^{\rho}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho\right)x_{i},$
(8)

where $u$ and $v$ are constants, and $x_{i}$ $\left(i=1,\,2,\,3\right)$
denotes Cartesian coordinates. The temperatures of the core, shell, and
background can be defined as $\varphi_{c}$, $\varphi_{s}$, and $\varphi_{b}$,
respectively. These temperatures can be obtained as follows:

$\left\\{\begin{array}[]{lll}\varphi_{c}=u_{c}x_{i}\\\
\varphi_{s}=\left(u_{s}+v_{s}\displaystyle\int_{\rho_{c}}^{\rho}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho\right)x_{i}\\\
\varphi_{b}=u_{b}x_{i}\end{array}\right.$ (9)

where $u_{c}$, $u_{s}$, and $v_{s}$ are determined by the boundary conditions.
The exterior surfaces of the core and shell are denoted by $\rho_{c}$ and
$\rho_{s}$, respectively. The boundary conditions require continuity of
temperatures and normal heat fluxes, leading to the following equations:

$\left\\{\begin{array}[]{llll}u_{c}=u_{s}\\\
u_{b}=u_{s}+v_{s}\displaystyle\int_{\rho_{c}}^{\rho_{s}}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho\\\
u_{c}=2v_{s}\kappa_{s}\left(\kappa_{c}-\kappa_{s}\right)^{-1}g\left(\rho_{c}\right)^{-1}\\\
u_{b}=2v_{s}\kappa_{s}\left(\kappa_{ei}-\kappa_{s}\right)^{-1}g\left(\rho_{s}\right)^{-1}\end{array}\right.$
(10)

Here, $\kappa_{ei}$ represents the effective thermal conductivity of the core-
shell structure along the direction of $x_{i}$. To obtain the expression for
$\kappa_{ei}$, one can solve Eq. (10). For this purpose, Xu et al. define the
semi-axis lengths of the core, $\lambda_{ci}$, and the shell, $\lambda_{si}$,
as follows:

$\left\\{\begin{array}[]{ll}\lambda_{ci}=\sqrt{\lambda_{i}^{2}+\rho_{c}}\\\
\lambda_{si}=\sqrt{\lambda_{i}^{2}+\rho_{s}}\end{array}\right.$ (11)

Here, $i=1,\,2,\,3$. Consequently, the volume fraction $f$ can be expressed
as:

$f=\frac{\lambda_{c1}\lambda_{c2}\lambda_{c3}}{\lambda_{s1}\lambda_{s2}\lambda_{s3}}=\frac{g\left(\rho_{c}\right)}{g\left(\rho_{s}\right)}$
(12)

Xu et al. also introduce the shape factor $d_{wi}$ along the direction of
$x_{i}$ as follows:

$d_{wi}=\frac{\lambda_{w1}\lambda_{w2}\lambda_{w3}}{2}\int_{0}^{\infty}\left(\tau+\lambda_{wi}^{2}\right)^{-1}\left(\left(\tau+\lambda_{w1}^{2}\right)\left(\tau+\lambda_{w2}^{2}\right)\left(\tau+\lambda_{w3}^{2}\right)\right)^{-1/2}d\tau,$
(13)

Here, the subscript $w$ can take the values $c$ or $s$, representing the shape
factor of the core or shell, respectively. Subsequently, they can obtain the
following expression:

$\begin{array}[]{cc}\displaystyle\int_{\rho_{c}}^{\rho_{s}}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho=\int_{\rho_{c}}^{\infty}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho-\int_{\rho_{s}}^{\infty}\left(\rho+\lambda_{i}^{2}\right)^{-1}g\left(\rho\right)^{-1}d\rho\\\
=2d_{ci}g\left(\rho_{c}\right)^{-1}-2d_{si}g\left(\rho_{s}\right)^{-1}.\end{array}$
(14)

Finally, Xu et al. derive a concise expression for $\kappa_{ei}$ as follows:

$\kappa_{ei}=\kappa_{s}\left[\frac{f\left(\kappa_{c}-\kappa_{s}\right)}{\kappa_{s}+\left(d_{ci}-fd_{si}\right)\left(\kappa_{c}-\kappa_{s}\right)}+1\right].$
(15)

The method described above is the standard approach for calculating the
Laplace equation. The shape factors satisfy the sum rule
$d_{w1}+d_{w2}+d_{w3}=1$. In principle, the effective thermal conductivity of
any core-shell structure can be obtained using Eq. (15) when the core-shell
structure is confocal or concentric. In fact, Eq. (15) can be reduced to
handle cylindrical (two-dimensional) cases by setting $\lambda_{w3}=\infty$,
resulting in $d_{w1}=\lambda_{w2}/\left(\lambda_{w1}+\lambda_{w2}\right)$,
$d_{w2}=\lambda_{w1}/\left(\lambda_{w1}+\lambda_{w2}\right)$, and $d_{w3}=0$
(the sum rule $d_{w1}+d_{w2}+d_{w3}=1$ is still satisfied).

Since $\gamma/\kappa$ is a constant, the effective radiative coefficient can
be expressed as:

$\gamma_{ei}=\gamma_{s}\left[\frac{f\left(\gamma_{c}-\gamma_{s}\right)}{\gamma_{s}+\left(d_{ci}-fd_{si}\right)\left(\gamma_{c}-\gamma_{s}\right)}+1\right],$
(16)

Here, $\gamma_{ei}$ represents the effective radiative coefficient of the
core-shell structure along the direction of $x_{i}$. By using Eqs. (15) and
(16), one can predict the effective thermal conductivity and effective
radiative coefficient. However, in order to achieve the same effect of
conduction and radiation, it is necessary to maintain $\gamma/\kappa$ as a
constant.

Figure 2: Steady results of thermal transparency. (a)-(c):
$\lambda_{c1}=\lambda_{c2}=2$ cm, $\lambda_{s1}=\lambda_{s2}=3$ cm,
$\kappa_{c}=2$ Wm-1K-1, $\beta_{c}=50$ m-1, $\kappa_{s}=0.62$ Wm-1K-1, and
$\beta_{s}=161.1$ m-1. (d)-(f): $\lambda_{c1}=2.5$ cm, $\lambda_{c2}=1.25$ cm,
$\lambda_{s1}=3$ cm, $\lambda_{s2}=2.08$ cm, $\kappa_{c}=0.5$ Wm-1K-1,
$\beta_{c}=200$ m-1, $\kappa_{s}=1.61$ Wm-1K-1, and $\beta_{s}=62$ m-1.
(g)-(i) are the references with pure background parameters. (j)-(l) are the
theoretical temperature distributions of the references. Circular (or
elliptical) dashed lines are plotted for the comparison with circular (or
elliptical) transparency. Adapted from Ref. [240]

Furthermore, to validate the theoretical analyses, Xu et al. conducted finite-
element simulations. The radiative emittance $j$ is shown in Fig. 1(d) to be
proportional to $T^{4}$ according to the Stefan-Boltzmann law. The conductive
flux $J_{\text{con}}$ depends on the temperature gradient, while the radiative
flux $J_{\text{rad}}$ is proportional to $T^{3}$, as depicted in Fig. 1(e).
These qualitative analyses indicate the significant role of thermal radiation
at high temperatures. Therefore, Xu et al. performed finite-element
simulations at three temperature intervals: (I) 273-313 K, representing a
small upper temperature limit where conduction (Con.) dominates; (II) 273-673
K, representing a medium upper temperature limit where conduction and
radiation (Rad.) are approximately equal; and (III) 273-4273 K, representing a
large upper temperature limit where radiation is the dominant mode of heat
transfer.

Figure 3: Steady simulations of thermal cloak. An inner object is coated by an
insulated layer with $\kappa=10^{-5}$ Wm-1K-1 and $\beta=10^{5}$ m-1. Since
the heat flux cannot enter into the insulated layer, the inner object plus the
insulated layer can be equivalently regarded as an insulated core with
$\kappa_{c}=10^{-5}$ Wm-1K-1 and $\beta_{c}=10^{5}$ m-1. Other parameters are
as follows. (a)-(c): $\lambda_{c1}=\lambda_{c2}=2.5$ cm,
$\lambda_{s1}=\lambda_{s2}=3$ cm, $\kappa_{s}=5.54$ Wm-1K-1, and
$\beta_{s}=18.1$ m-1. (d)-(f): $\lambda_{c1}=2.5$ cm, $\lambda_{c2}=1.25$ cm,
$\lambda_{s1}=3$ cm, $\lambda_{s2}=2.08$ cm, $\kappa_{s}=2.35$ Wm-1K-1, and
$\beta_{s}=42.5$ m-1. Adapted from Ref. [240]

Thermal transparency involves the design of a shell tailored to the object,
ensuring an undistorted temperature profile outside the shell, as depicted in
Fig. 2. By selecting appropriate parameters based on Eqs. (15) and (16), Xu et
al. demonstrated that the temperature profile outside the shell remains
undistorted, as shown in Figs. 2(a)-2(c) or Figs. 2(d)-2(f). Consequently, the
core-shell structure at the center becomes indistinguishable, as illustrated
in Figs. 2(g)-2(i). Figs. 2(j)-2(l) present the corresponding theoretical
results from the references using the same parameters as Figs. 2(g)-2(i). The
matching temperature profiles in both simulations and theory validate the
theoretical analyses.

Furthermore, Xu et al. demonstrated that with a small upper temperature limit
where conduction dominates, the temperature gradient outside the shell remains
nearly uniform, as depicted in the first row of Fig. 2. As the upper
temperature limit increases, the effect of radiation becomes prominent,
resulting in nonuniform temperature gradients outside the shell, as shown in
the last two rows of Fig. 2.

A thermal cloak is capable of shielding any object within it from detection.
Generally, an insulated layer is employed to prevent heat flux from reaching
the object. Consequently, the object along with the insulated layer can be
treated as an insulated core, where $\kappa_{c}=\gamma=0$. Moreover, Xu et al.
designed a shell based on Eqs. (15) and (16) to eliminate the influence of the
insulated core. The simulation results are presented in Figs. 3(a)-3(c) and
Figs. 3(d)-3(f). Evidently, the isotherms remain separated from the object,
indicating that the heat flux is unable to enter the object. Additionally, the
temperature profiles of the background remain undistorted. This successful
outcome demonstrates the expected cloaking effect.

Figure 4: Steady simulations of thermal expander. The sizes are
$\lambda_{c1}=2.08$ cm, $\lambda_{c2}=4.17$ cm, $\lambda_{s1}=3.46$ cm,
$\lambda_{s2}=5$ cm, and the width between hot and cold sources is 6 cm. Other
parameters are as follows. (a)-(c): $\kappa_{s}=4.91$ Wm-1K-1 and
$\beta_{s}=20.3$ m-1. (d)-(f): pure background parameters. Adapted from Ref.
[240]

The thermal expander concept aims to magnify a small heat source into a larger
one by utilizing the design of two elliptical cloaks. To demonstrate this
effect, Xu et al. assembled two elliptical cloaks together and extracted a
quarter of the entire structure as the expander, as depicted in Fig. 1(c).
According to the uniqueness theorem in thermotics, the temperature
distribution of the background remains undistorted, thereby achieving the
desired expander effect. Finite-element simulations were conducted and
presented in Figs. 4(a)-4(c). Clearly, the isotherms of the background appear
as straight lines, indicating the excellent performance of the proposed
structure. For comparison, Xu et al. also provided simulation results for a
pure background material. These results reveal that the isotherms of the
background become distorted, as shown in Figs. 4(d)-4(f).

Figure 5: Transient simulations of transparency, cloak, and expander. The
sizes and material parameters of (a)-(c), (d)-(f), and (g)-(i) are the same as
those for Figs. 2(d)-2(f), Figs. 3(d)-3(f), and Figs. 4(a)-4(c), respectively.
The density and heat capacity of the background are $\rho C=10^{6}$ Jm-3K-1.
Other parameters are as follows. (a)-(c): $\left(\rho C\right)_{c}=5\times
10^{5}$ Jm-3K-1 and $\left(\rho C\right)_{s}=1.61\times 10^{6}$ Jm-3K-1.
(d)-(f): $\left(\rho C\right)_{s}=2.35\times 10^{6}$ Jm-3K-1. (g)-(i):
$\left(\rho C\right)_{s}=5\times 10^{5}$ Jm-3K-1. Adapted from Ref. [240]

Xu et al. demonstrated that the theoretical analyses are applicable not only
to steady states but also to transient cases. To illustrate this point, Xu et
al. considered density and heat capacity. In order to design transient
transparency and cloak, the value of heat diffusivity $\kappa/\left(\rho
C\right)$ was set as a constant. Based on the simulation results, the
performance of this approach remained satisfactory. The results at
$t=10,\,20,\,60$ mins are depicted in Figs. 5(a)-5(c) and Figs. 5(d)-5(f),
respectively.

To showcase the effect of transient expansion and achieve the optimal
transient effect, Xu et al. employed an optimization method and set the
diffusivity of the shell to be larger than that of the background. The results
at $t=6,\,10,\,20$ mins are presented in Figs. 5(g)-5(i).

##  3 Potential applications of radiative metamaterials: thermal camouflage
and radiative cooler

The development of diffusion metamaterials for radiation control holds
significant importance for our future lives, owing to its potential
applications such as thermal camouflage [245, 246, 247, 248, 249, 250, 251,
252] and radiative cooling. Thermal camouflage refers to a device that not
only prevents the detection of objects but also generates deceptive signals.
Researchers have designed systems that can mislead the detection of cloaked
objects. Additionally, encrypted thermal printing has been achieved, enabling
object detection only when an appropriate heat source is applied. While the
original structure for thermal camouflage is two-dimensional, there are also
studies focusing on realizing the same functionality using three-dimensional
structures.

Moreover, radiative cooling is another application field worth mentioning.
Radiative cooling involves the automatic achievement of cooling effects
without the need for an external source. The concept was first proposed in
1978, but at that time, it remained primarily theoretical. It was in 2014 that
a practical radiative cooler was designed based on the photonic approach. The
device operates by selectively reflecting electromagnetic waves in the mid-
infrared range. Since then, researchers have developed more efficient methods
to achieve the same functionality. Experimental results have also confirmed
the effectiveness of these devices. We believe that in the future, radiative
coolers could find widespread use in our daily lives, provided that the costs
are reduced to a certain level.

##  4 Outlook: radiative metamaterials from microscopic view

Over the past decades, research on diffusion metamaterials has undergone
significant changes. Initially, the focus was on single-function devices, but
it has since shifted towards multi-functionality [253, 254]. Similarly, there
has been a transition from linear to nonlinear response [255, 256, 257, 258,
259, 260], from temperature-independent to temperature-dependent devices [261,
262, 263], from single thermal to thermoelectrical effects [264, 265, 266,
267], from spatial to spatiotemporal metamaterials [268, 269, 270, 271, 272],
and from temperature diffusion to temperature waves [273, 274, 275, 276, 277].
Furthermore, the research methodology has been applied to other fields as
well, including hydrodynamics [278, 279, 280, 281, 282, 283, 284], plasma
physics [285], mass diffusion [286, 287], and topology [288]. Novel concepts
such as programmable metamaterials [289, 290], nonreciprocity metamaterials
[291], intelligent metamaterials [292, 293, 294, 295, 296, 297, 298, 299],
dipole-assisted thermotics [300], and negative thermal transport [301] have
also been proposed. Additionally, the research on networks [302, 303] and
phase transitions [304, 305, 306, 307] is expected to make further progress.
In the future, the impact of diffusion metamaterials on the development of
thermal diodes [308, 309, 310, 311, 312] is eagerly anticipated. Machine
learning [313, 314] is also increasingly being incorporated into the research
methodology. While significant progress has been made in the study of
diffusion metamaterials for conduction and radiation, the research has
primarily focused on macroscopic scales. Exploring the same effects at the
microscopic scale is an intriguing topic [315]. To validate the Fourier law
from a microscopic perspective, researchers have designed various models. The
simplest among them is the harmonic lattice model [316, 317, 318, 319, 320,
321, 322, 323, 324, 325, 326, 327, 328, 329], which includes one-dimensional
ordered cases [330], higher-dimensional ordered cases, one-dimensional
disordered cases [331, 332, 333, 334, 335, 336, 337, 338, 339, 340], and two-
dimensional disordered cases [341, 342, 343, 344, 345, 347, 348, 346].
Additionally, the harmonic lattices with self-consistent reservoirs have been
studied [349, 350, 351, 352, 353, 354]. More complex models involve
interacting systems [355, 356, 357, 358, 359, 360]. The research in this field
can be divided into two parts: momentum-conserving models, such as the FPU
model [361, 362, 363, 364, 365, 366, 367], and momentum-non-conserving models
[368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382].
Some of these models provide essential insights into illustrating the Fourier
law. Therefore, exploring the possibility of achieving novel functions such as
cloaking and concentration in a similar manner is a worthwhile topic to
explore. Furthermore, the experimental validation of microscopic theories
poses a significant challenge. However, advancements in nanotechnology inspire
us to move forward, and the combination of diffusion metamaterials with low-
dimensional systems offers a new research field.

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