The Equivalent Thermal Conductivity of the Micro/Nano Scaled Periodic Cubic Frame Silver and Its Thermal Radiation Mechanism Analysis

: Currently, there are few studies on the inﬂuence of microscale thermal radiation on the equivalent thermal conductivity of microscale porous metal. Therefore, this paper calculated the equivalent thermal conductivity of high-porosity periodic cubic silver frame structures with cell size from 100 nm to 100 µ m by using the microscale radiation method. Then, the media radiation characteristics, absorptivity, reﬂectivity and transmissivity were discussed to explain the phenomenon of the radiative thermal conductivity changes. Furthermore, combined with spectral radiation properties at the different cross-sections and wavelength, the radiative transmission mechanism inside high-porosity periodic cubic frame silver structures was obtained. The results showed that the smaller the cell size, the greater radiative contribution in total equivalent thermal conductivity. Periodic cubic silver frames ﬂuctuate more in the visible band and have better thermal radiation modulation properties in the near infrared band, which is formed by the Surface Plasmon Polariton and Magnetic Polaritons resonance jointly. This work provides design guidance for the application of this kind of periodic microporous metal in the ﬁeld of thermal utilization and management.


Introduction
Since the discovery of microscale thermal radiation effects by Tian et al. [1,2] in the 1960s, these phenomena have attracted scholars' attention worldwide. Particularly in recent years, with the increasing maturity of the microscale processing and measurement technology, whose thermal radiation properties and modulation methods in the direction of solar radiation, photoelectric conversion, infrared detection, optical imaging, infrared stealth, etc. have become a research hotspot with a highly practical application value. Within the microscale, the heat transfer between two or more close objects can far exceed blackbody radiation, where the Planck blackbody radiation law cannot function anymore. Therefore, the explanation of the heat transfer phenomenon of macroscopic objects is no longer applicable for the object whose characteristic size is comparable to or smaller than the wavelength of thermal radiation [3]. In comparison to macroscopic radiation, the phenomena like Surface Plasmon Polaritons (SPPs) [4][5][6], Phonon Polariton Effect [7,8], Photonic Band-gap Effect [9], Photon Tunneling Effect [10], etc. produced in the field of microscale thermal radiation.
Various scholars have focused on the research of the thermal radiation properties of periodic microstructures. Lee et al. [11,12] calculated the emissivity of silver structures using rigorous coupled-wave analysis and found that the emissivity can reach large emission peaks at resonant frequencies with excited magnetic polarization. Wang et al. [13] investigated the effect of complex grating structures on thermal radiation properties using a timedomain finite-difference method. Qiu et al. [14] researched thermal radiation properties of random rough surfaces. Huang et al. [15,16] analyzed the thermal radiation properties

Mathematical Methods
Finite element method (FEM) [21][22][23] as a highly adaptable numerical method used wildly in the calculation of thermal radiation properties of microstructures. By dividing a continuous three-dimensional region into a finite number of small regions, FEM applied to any physical field described by differential equations theoretically, based on the variational principle. The following work was carried out by using FEM.

Absorptivity, Reflectivity and Transmissivity
The research of microscale thermal radiation is generally presented in terms of electromagnetic theory [24], with describing the distribution of electromagnetic field by Maxwell equations in vacuum as [24]: where B indicates the magnetic flux density, Wb/m 2 ; D stands for the electric flux density while D e is the electric displacement vector, C/m 2 ; E presents the electric field, V/m; H is the magnetic field, A/m; J indicates the current density, A/m 2 ; t is the time, s, ρ is the volume charge density, C/m 3 . To obtain the Maxwell equations in media instead of vacuum, the material equations are needed as [24]: where E 0 , H 0 , E i+1 , and H i+1 , respectively represent electromagnetic fields at the outer surface of the multilayer films. When the electromagnetic wave is incident vertically, the reflection coefficient r and transmission coefficient t of the multilayer film structure are [26]: r = (m 11 + m 12 n t+1 )n 0 − (m 21 + m 22 n t+1 ) (m 11 + m 12 n t+1 )n 0 + (m 21 + m 22 n t+1 ) (10) t = 2n 0 (m 11 + m 12 n t+1 )n 0 + (m 21 + m 22 n t+1 ) (11)  The α A , α R and α T can be further obtained from the reflectance and transmittance as follows [26]:

The Equivalent Thermal Conductivity
Based on Equation (4), the spectral absorption coefficient β α and a spectral scattering coefficient β s can be calculated as follows [27]: where A is the unit cross-sectional area and N is the number of particles per unit volume. The spectral extinction coefficient β e can be calculated as follows [27]: According to the optical thickness approximation, the equivalent radiative thermal transfer equation is as follows [27]: where σ SB is the Stefan-Boltzmann constant, κ rad is the radiative equivalent thermal conductivity and β e,R is the Rosseland average extinction coefficient, which can be calculated as [24]: 1 where λ is the wavelength, f (λ,T) is the spectral distribution of Planck blackbody emission, given by [24]: where e b,λ is the blackbody spectral intensity, C 1 = 2πhc 2 is the first radiation constant, C 2 = hc/k B is the second radiation constant and h is the Planck constant [28]. Therefore, Equation (16) would be used to calculate the κ rad in the following calculation. According to Fourier's law, the governing equations of the thermal conduction progress of a three-dimensional object can be established during the transient temperature field T(x, y, z, t) as [27,29]: where κ x , κ y , and κ z are the thermal conductivities of the object along the x, y and z directions, all three values of which are equal to κ s for isotropic materials, A is the area. In this research, the thermal conductivities of pure Ag at room temperature (T = 300 K) were adopted from Young's [30] experimental measurements, that is κ s = 406 W/(m·K); ρ is the density of the object kg/m 3 ; c is the specific heat capacity of the object J/(kg·K) and q is the density of the thermal source in the object, W/m 2 . According to Fourier's law, for the heat flux q n through differential element, the Equation (20) is obtained [29]: (20) where κ cond is the equivalent conductive thermal conductivity, n presents the heat transfer direction. It should note that, the thermal conduction progress without consider radiation. The total equivalent thermal conductivity κ total is obtained by [28]: Therefore, the Equation (21) would be used to calculate the κ total in the following calculation.

Physical Model
In this paper, the three-dimensional geometric model established with simplified cubic hollow as shown in Figure 1. Silver was selected as the material of the infinite plate and periodic cubic structure for its good physical properties, with regardless of the oxidation condition and the effect of the changes in the n, k value of the material. The cubic skeleton material chosen is metallic silver which is extensively used in photonic crystals. In consideration of the effect of wavelength on n, k, this curve was obtained by interpolating the experimental points measured by Yang et al. [31] using Bessel spline function as shown in Figure 2.
where κx, κy, and κz are the thermal conductivities of the object along the x, y and z directions, all three values of which are equal to κs for isotropic materials, A' is the area. In this research, the thermal conductivities of pure Ag at room temperature (T = 300 K) were adopted from Young's [30] experimental measurements, that is κs = 406 W/(m·K); ρ is the density of the object kg/m 3 ; c is the specific heat capacity of the object J/(kg·K) and q is the density of the thermal source in the object, W/m 2 . According to Fourier's law, for the heat flux qn through differential element, the Equation (20) is obtained [29]: where κcond is the equivalent conductive thermal conductivity, n presents the heat transfer direction. It should note that, the thermal conduction progress without consider radiation. The total equivalent thermal conductivity κtotal is obtained by [28]: Therefore, the Equation (21) would be used to calculate the κtotal in the following calculation.

Physical Model
In this paper, the three-dimensional geometric model established with simplified cubic hollow as shown in Figure 1. Silver was selected as the material of the infinite plate and periodic cubic structure for its good physical properties, with regardless of the oxidation condition and the effect of the changes in the n, k value of the material. The cubic skeleton material chosen is metallic silver which is extensively used in photonic crystals. In consideration of the effect of wavelength on n, k, this curve was obtained by interpolating the experimental points measured by Yang et al. [31] using Bessel spline function as shown in Figure 2.

Calculation Domain of Infinity Silver Plate
In order to verify the numerical simulation method, the infinite silver plate with thickness of 500 nm is calculated using the transmission matrix and finite element method, respectively, and the absorption curves are plotted using the obtained results as shown in Figure 3. In this section, Transverse Electromagnetic Wave (TEW) is used for vertical incidence. Due to the excitation effect of metals at far-infrared wavelengths at the mi-

Calculation Domain of Infinity Silver Plate
In order to verify the numerical simulation method, the infinite silver plate with thickness of 500 nm is calculated using the transmission matrix and finite element method, respectively, and the absorption curves are plotted using the obtained results as shown in Figure 3. In this section, Transverse Electromagnetic Wave (TEW) is used for vertical incidence. Due to the excitation effect of metals at far-infrared wavelengths at the microscale, it is impossible to obtain a more regular modulated wavelength range, therefore the wavelength range is taken as visible and mid-infrared wavelengths, which are 300 nm-3 µm.

Calculation Domain of Infinity Silver Plate
In order to verify the numerical simulation method, the infinite silver plate with thickness of 500 nm is calculated using the transmission matrix and finite element method respectively, and the absorption curves are plotted using the obtained results as shown in Figure 3. In this section, Transverse Electromagnetic Wave (TEW) is used for vertical in cidence. Due to the excitation effect of metals at far-infrared wavelengths at the mi croscale, it is impossible to obtain a more regular modulated wavelength range, therefore the wavelength range is taken as visible and mid-infrared wavelengths, which are 300 nm-3 µm. As shown in Figure 3, the calculated results using FEM almost overlap with the ab sorptance curve obtained by the Transfer Matrix method (TMM), which is almost an exac analytical solution. The absorptivity increases with wavelength and then decreases sharply, with a maximum peak at 0.32 µm, with a maximum absorptivity of 0.98, which indicates that the absorption capacity of the flat plate increases with wavelength and then decreases sharply. It is consistent with the trend that the imaginary part of the complex permittivity of Ag first decreases and then increases and reaches a minimum at 0.32 µm as shown in Figure 2. Since the two curves are almost the same, the FEM calculation resul is consistent with the analytical solution. Therefore, the following calculations are all car ried out using the finite element method. As shown in Figure 3, the calculated results using FEM almost overlap with the absorptance curve obtained by the Transfer Matrix method (TMM), which is almost an exact analytical solution. The absorptivity increases with wavelength and then decreases sharply, with a maximum peak at 0.32 µm, with a maximum absorptivity of 0.98, which indicates that the absorption capacity of the flat plate increases with wavelength and then decreases sharply. It is consistent with the trend that the imaginary part of the complex permittivity of Ag first decreases and then increases and reaches a minimum at 0.32 µm, as shown in Figure 2. Since the two curves are almost the same, the FEM calculation result is consistent with the analytical solution. Therefore, the following calculations are all carried out using the finite element method.

Computational Domain of the Periodic Cubic Structure
As shown in Figure 4, the computational domain of the periodic cubic is taken to be unit cubic connected structure with unit cubic cell length a and an internal cubic pore cell length b. Therefore, the effective cell size can be calculated by d p = [(18 ab 2 − 12 b 3 )/π] 1/3 and the porosity can be obtained by ϕ = (3 ab 2 − 2 b 3 )/a 3 . The incident surface to the top of the cubic is set at a distance h = 250 nm, with periodic boundary conditions on all surfaces except the top and bottom, respectively, which are set as the emitting and receiving ports, where the incident magnetic field strength H = 1 A/m is given, plus a perfect absorption layer. Furthermore, the integral calculation band of thermal radiation was 2.6 µm < λ < 90 µm. The incidence angles were divided into 20,000-unit angles from 0 to 90 • , and the frequency interval was 2 × 10 12 Hz, which was used in the following calculations to ensure the calculation accuracy.
cubic is set at a distance h = 250 nm, with periodic boundary conditions on all surfaces except the top and bottom, respectively, which are set as the emitting and receiving ports, where the incident magnetic field strength H = 1 A/m is given, plus a perfect absorption layer. Furthermore, the integral calculation band of thermal radiation was 2.6 µm < λ < 90 µm. The incidence angles were divided into 20,000-unit angles from 0 to 90°, and the frequency interval was 2 × 10 12 Hz, which was used in the following calculations to ensure the calculation accuracy. band is plotted as well as the error as shown in Figure 5. It can be obtained from Figure 5, as the value of Me increases, the absorbance curve keeps changing. When Me ≥ 5, the change of the absorption curve is no longer obvious with the increase of Me. Therefore, the smallest error is obtained when the Me ≥ 5, which is the domain dividing into 740,000 hexahedral elements at least. As a result, the domain would be divided into around 740,000 elements in the following calculations to ensure smaller errors. In order to verify the independence of the computational model, a dimensionless number Me is defined, Me = λ/δ 1/2 , where λ denotes the incident wave wavelength and δ denotes the characteristic length of elements the unit calculated domain. By adjusting the δ, the absorption with wavelength for different accuracy pairs in the visible wavelength band is plotted as well as the error as shown in Figure 5. It can be obtained from Figure 5, as the value of Me increases, the absorbance curve keeps changing. When Me ≥ 5, the change of the absorption curve is no longer obvious with the increase of Me. Therefore, the smallest error is obtained when the Me ≥ 5, which is the domain dividing into 740,000 hexahedral elements at least. As a result, the domain would be divided into around 740,000 elements in the following calculations to ensure smaller errors.

The Equivalent Thermal Conductivity
In order to discuss the thermal radiation proportion in the total heat transfer proces more intuitively, the equivalent thermal conductivity was chosen in this section as th evaluation index to analyze the changes of the three kinds of equivalent thermal conduc

The Equivalent Thermal Conductivity
In order to discuss the thermal radiation proportion in the total heat transfer process more intuitively, the equivalent thermal conductivity was chosen in this section as the evaluation index to analyze the changes of the three kinds of equivalent thermal conductivities with the cell size and porosity. Four group of higher porosities models ϕ = 0.896, ϕ = 0.784, ϕ = 0.684 and ϕ = 0.500 were calculated at T = 300 K, respectively, with the temperature interval was selected as 1 K. In addition, the integral calculation band of thermal radiation was 2.6 µm < λ < 90 µm. The thermal conductivity of pure Ag was set as κ s = 406 W/m·K followed Yong's [29] research. The spectral refractive index n and spectral extinction coefficient k were adopted from Yang's experimental measurements [31], as seen in Figure 2. The κ total , κ cond and κ rad were calculated from Equations (12) and (15), respectively as a function of the size of the structure, as shown in Figure 6.

The Equivalent Thermal Conductivity
In order to discuss the thermal radiation proportion in the total heat transfer process more intuitively, the equivalent thermal conductivity was chosen in this section as the evaluation index to analyze the changes of the three kinds of equivalent thermal conductivities with the cell size and porosity. Four group of higher porosities models φ = 0.896, φ = 0.784, φ = 0.684 and φ = 0.500 were calculated at T = 300 K, respectively, with the temperature interval was selected as 1 K. In addition, the integral calculation band of thermal radiation was 2.6 µm < λ < 90 µm. The thermal conductivity of pure Ag was set as κs = 406 W/m·K followed Yong's [29] research. The spectral refractive index n and spectral extinction coefficient k were adopted from Yang's experimental measurements [31], as seen in Figure 2. The κtotal, κcond and κrad were calculated from Equations (12) and (15), respectively as a function of the size of the structure, as shown in Figure 6. As shown in Figure 6, the κcond is not significantly impacted by changes in cell size dp, while the κcond increased with the φ decreased. This is because as the φ decreases, the proportion of the gas phase thermal conductivity increases, which ultimately leads to a decrease in the overall κcond. For models φ = 0.784, φ = 0.684 and φ = 0.500, as the structure size increases, the thermal radiation and total heat transfer decrease first and then increase, while at φ = 0.896, as the structure size increases, the thermal radiation and total heat transfer decrease first and then increase to a peak before decreasing. It is because when the aperture size is comparable to the peak Planck blackbody radiation, the microscale effect is enhanced so that the total extinction coefficient decreases, resulting in the peak in Figure 6a. When a < 1 µm, the larger the porosity for the same size, the larger the share of the κtotal in the κrad, and the same trend of change in the κrad and the κtotal. Meanwhile, when φ = 0.896, dp < 1 µm, the κrad plays a dominant role in the κtotal. Hence, on the basis of this set of models, the next section shall concern the reasons for this phenomenon from the perspective of the thermal radiation medium properties. As shown in Figure 6, the κ cond is not significantly impacted by changes in cell size d p , while the κ cond increased with the ϕ decreased. This is because as the ϕ decreases, the proportion of the gas phase thermal conductivity increases, which ultimately leads to a decrease in the overall κ cond . For models ϕ = 0.784, ϕ = 0.684 and ϕ = 0.500, as the structure size increases, the thermal radiation and total heat transfer decrease first and then increase, while at ϕ = 0.896, as the structure size increases, the thermal radiation and total heat transfer decrease first and then increase to a peak before decreasing. It is because when the aperture size is comparable to the peak Planck blackbody radiation, the microscale effect is enhanced so that the total extinction coefficient decreases, resulting in the peak in Figure 6a. When a < 1 µm, the larger the porosity for the same size, the larger the share of the κ total in the κ rad , and the same trend of change in the κ rad and the κ total . Meanwhile, Energies 2021, 14, 4158 9 of 15 when ϕ = 0.896, d p < 1 µm, the κ rad plays a dominant role in the κ total . Hence, on the basis of this set of models, the next section shall concern the reasons for this phenomenon from the perspective of the thermal radiation medium properties.

Absorptivity, Reflectivity and Transmissivity
In order to research the phenomenon of the large share of κ rad at micro/nano sizes, a model with cell length a = 500 nm, an internal control cell length of b = 400 nm and porosity ϕ = 0.896 is taken as an example for this paper, and the analysis of the thermal radiation properties of the medium at wavelengths comparable to the size of the structure is carried out specifically.
Firstly, the section calculates the absorptivity, reflectivity and transmissivity of infinitely large flat and periodic high porosity cubic frame structures at wavelengths in the visible and near infrared, as shown in Figure 7. For structures with finite large flat plates are shown in Figure 7a. When the wavelength is smaller than the size of the structure in the propagation direction, the reflectivity is lower and the absorption is the first to increase until it plummets close to 1. However, when the wavelength is larger than the size of the structure in the propagation direction, the absorption decreases to close to 0, at which point the reflectivity approaches 1. The transmissivity is always close to 0, and the curves of absorption and reflectivity level off when the wavelength is two times larger than the size in the propagation direction. When the wavelength approaches the size of the structure in the direction of propagation, the absorption decreases abruptly, and the reflectivity increases abruptly. This is close to the distribution curve of the real part of the dielectric constant of metallic silver, and in the infrared band the dielectric constant of silver is greater than 1, close to total reflection, which is consistent with a reflectivity close to 1 in the infrared band. As shown in Figure 7b, the reflection, absorption and transmission rates of the peri odic high porosity cubic frame structure in the visible and mid-infrared wavelengths dif fer significantly from those of an infinite flat plate of the same thickness. At visible wave lengths, the oscillation of the periodic cubic frame structure with a structure size of 500 nm is more severe, while its absorption, reflection and transmission rates do not show monotonicity in their variation with wavelength. At wavelengths greater than 2 µm, the absorption, reflection and transmission rates of this structure are close to those of the fla plate structure. However, the absorption of the structure increases first and then decreases in the near infrared band, with a maximum value of 0.178 at 1.425 µm wavelength, after which it decreases to a level close to 0.05. Although this absorption is not high, it is a significant modulation of the absorption compared to the zero absorption of the flat struc As shown in Figure 7b, the reflection, absorption and transmission rates of the periodic high porosity cubic frame structure in the visible and mid-infrared wavelengths differ significantly from those of an infinite flat plate of the same thickness. At visible wavelengths, the oscillation of the periodic cubic frame structure with a structure size of 500 nm is more severe, while its absorption, reflection and transmission rates do not show monotonicity in their variation with wavelength. At wavelengths greater than 2 µm, the absorption, reflection and transmission rates of this structure are close to those of the flat plate structure. However, the absorption of the structure increases first and then decreases in the near infrared band, with a maximum value of 0.178 at 1.425 µm wavelength, after which it decreases to a level close to 0.05. Although this absorption is not high, it is a significant modulation of the absorption compared to the zero absorption of the flat structure in this band. Meanwhile, the peaks and troughs of the reflectivity and transmissivity of the structure in the NIR band correspond to the same wavelength. The reflectivity has two troughs at 0.7 µm and 1.28 µm, where the reflectivity is nearly 0. The transmissivity has two peaks, which correspond to a transmissivity of nearly 0.9, respectively. The reflectivity peaks coincide with the transmissivity peaks, where the reflectivity reaches a maximum of 0.4 at a wavelength of 0.96 µm, and the transmissivity is at a minimum of 0.525. In order to better investigate the NIR band modulation properties of periodic high porosity microstructures, the following is a mechanistically sound explanation of the spectral radiation properties at the reflectivity and absorptivity peaks in this band, respectively.

Spectral Radiation Properties at the Reflectivity Peak
In order to clearly illustrate the electric field direction and magnetic field strength, at different locations of cross sections within the calculated cell at the reflectivity and absorptivity peaks, cross sections are now taken at Y = 0 nm, Y = 100 nm, Y = 200 nm, Y = 225 nm and Y = 250 nm, respectively, as shown in Figure 4, with the centre of the cell as the origin, along the vertical one magnetic field direction. Where the black wire frame indicates the plane obtained by targeting the plane to the material, that is the area within the frame indicates the magnetic and electric field distribution within the material.
It can be seen from Figure 8, the distribution of the electric field direction and magnetic field intensity for each cross-section at the peak of absorptivity, which is at a wavelength of 0.96 µm. Where the direction of electric field transmission inside the cubic frame structure changes at the cross-section Y = 0 nm and Y = 100 nm cross sections. Due to the coupling of the induced magnetic field formed by induction with the incident magnetic field an inverse magnetic resonance is formed, this resonance is a change in the direction of the electric field transmission inside. Due to the small size of the structure a closed circuit cannot be formed, but the induced magnetic field formed by it weakens the magnetic field there; thus the magnetic field strength at the four cubic sections is lower. The phenomenon is most pronounced in the section above the structure. At the Y = 200 nm cross-section, this plane is just above the surface of the cubic frame structure. As shown in Figure 8c, the direction of the electric field near the surface of the prism is deflected and the resonance can be observed at the surface of the prismatic structure. It decays rapidly in the interior, which is typical Surface Plasmon Polariton (SPP). The SPP resonance is generated by the incident magnetic field coupled to a locally induced electric field and shows an exponential decay of the surface swift electromagnetic waves propagating along the interface between the free space and the structure and forming a resonance with the plasma excitations on the surface of the silver cubic frame structure. In comparison with Figure 8c-e, it can be observed that the SPP resonance is more pronounced the closer the central section of the cubic cell.

Spectral Radiation Properties at the Absorptivity Peak
In order to study the peak absorptivity of the cubic frame structure, cross sections were taken at positions Y = 0 nm, Y = 100 nm, Y = 200 nm, Y = 225 nm and Y = 250 nm at a wavelength of 1.425 µm as shown in Figure 3, respectively, and the logarithmic distribution of the electric field direction and magnetic field intensity of each cross section was investigated, and cloud plots were made.
As shown in Figure 9, the distribution of the electromagnetic field at each cross-section is broadly similar to Figure 8, where the surface plasma resonance effect triggers an enhancement and weakening of the induced magnetic field, with a significant attenuation at the centre along the propagation direction compared to Figure 8. When the electromagnetic wave is incident on the surface of the periodic cubic frame, it is known from the law of flutters that a time-varying electromagnetic wave will form an induced current inside the cavity. This current subsequently generates an induced magnetic field in the opposite di-rection, which couples with the incident magnetic field to form an anti-magnetic resonance. This resonance is known as the and Magnetic Polaritons (MPs) resonance, which forms a parallel electric field direction current inside the internal cavity. Meanwhile, comparing the absorption and reflection peaks in Figure 7, it can be seen that the absorption peak at a wavelength of 1.425 µm is wider, which is typical of the magnetic polarization excitation phenomenon. Moreover, the excitation phenomenon does not change with the angle of incidence, which is of great importance in industrial production. netic field there; thus the magnetic field strength at the four cubic sections is lower. The phenomenon is most pronounced in the section above the structure. At the Y = 200 nm cross-section, this plane is just above the surface of the cubic frame structure. As shown in Figure 8c, the direction of the electric field near the surface of the prism is deflected and the resonance can be observed at the surface of the prismatic structure. It decays rapidly in the interior, which is typical Surface Plasmon Polariton (SPP). The SPP resonance is generated by the incident magnetic field coupled to a locally induced electric field and shows an exponential decay of the surface swift electromagnetic waves propagating along the interface between the free space and the structure and forming a resonance with the plasma excitations on the surface of the silver cubic frame structure. In comparison with Figure 8c-e, it can be observed that the SPP resonance is more pronounced the closer the central section of the cubic cell.  nance. This resonance is known as the and Magnetic Polaritons (MPs) resonance, which forms a parallel electric field direction current inside the internal cavity. Meanwhile, comparing the absorption and reflection peaks in Figure 7, it can be seen that the absorption peak at a wavelength of 1.425 µm is wider, which is typical of the magnetic polarization excitation phenomenon. Moreover, the excitation phenomenon does not change with the angle of incidence, which is of great importance in industrial production.

Conclusions
This paper has calculated the equivalent thermal conductivity of periodic cubic frame silver with porosities of 0.896, 0.784, 0.684 and 0.500, respectively and cell size from 100 nm to 100 µm using the finite element method. In order to explain the effect of microscale effects on the equivalent radiative thermal conductivity, this work further investigated the reflectivity, absorptivity and transmissivity of periodic cubic frame silver at the incident band of 300 nm-3 µm. Then, the magnetic field strength and electric field direction were analyzed at different cross sections in the band at the peak of absorptivity and reflectivity, in order to explaining the thermal radiation transmission mechanism. The conclusions are consulted as follows: 1. In the equivalent thermal conductivity of a periodic cubic frame structure, the smaller the cell size of the structure, the larger the proportion of thermal radiation. When the

Conclusions
This paper has calculated the equivalent thermal conductivity of periodic cubic frame silver with porosities of 0.896, 0.784, 0.684 and 0.500, respectively and cell size from 100 nm to 100 µm using the finite element method. In order to explain the effect of microscale effects on the equivalent radiative thermal conductivity, this work further investigated the reflectivity, absorptivity and transmissivity of periodic cubic frame silver at the incident band of 300 nm-3 µm. Then, the magnetic field strength and electric field direction were analyzed at different cross sections in the band at the peak of absorptivity and reflectivity, in order to explaining the thermal radiation transmission mechanism. The conclusions are consulted as follows: