Nowadays, research attention have gradually been turned to the nano-scale since micro-electro-mechanical systems (MEMS) developed rapidly. Nano-scale heat transfer including the near-field radiation has been more and more concerned with.^[1] Thermal radiation on a nano-scale, i.e., near-field radiation, is different from macroscale thermal radiation, and has size effect.

Early in 1971, Polder and Hove^[2] initially deduced the radiative heat flux between two parallel plates with micro-scale gap. Since then, numerous studies on the near-field radiative heat transfer between two semi-infinite bodies, ^{[3– 5]} between a spherical particle and a semi-infinite body^{[6, 7]} or between two spherical particles^{[8– 10]} have been carried out.

It has been investigated that as the gap between the two mediums is so small that it is comparable to or shorter than the characteristic wavelength of thermal radiation (λ _T), Stefan– Boltzmann law is no longer applicable. The radiation will have wave interference effect and photon tunneling effect, ^{[11– 13]} which greatly enhances the radiative heat transfer. The near-field radiative heat transfer can exceed blackbody radiation by 5– 6 orders of magnitude.^{[14– 16]}

Nanoscale radiative heat transfer promotes the research and development of near-field radiation.^[17] For a mesoporous material, the near-field radiation across the pore cannot be ignored, which is rarely reported. Although the thermal property of amorphous mesoporous material has caused great attention, ^{[18, 19]} the heat transfer mechanism for amorphous mesoporous material is not very clear yet. Therefore, the research of thermal conductivity of amorphous mesoporous material is still a hot point. For amorphous mesoporous alumina film (AAO template) as an amorphous material, its thermal conductivity has been rarely reported relatively.^{[20, 21]}

In this paper, we focus on the near-field radiation across a cylindrical pore by employing the fluctuation dissipation theorem and Green function, and its influence on the thermal conductivity of the mesoporous alumina. Porosity weighted dilute medium (PWDM) model^[22] is used to combine the near-field radiation with the thermal conductivities of gas confined in the pore of mesoporous alumina substrate. The combined thermal conductivity of the mesoporous silica is obtained and compared with the measurement. Finally, the effects of the pore size and temperature are also investigated.

Near-field radiative heat transfer across a pore in mesoporous alumina is analyzed. The schematic nanostructure of mesoporous alumina is shown in Fig.  1(a), which consists of uniformly distributed, unconnected cylindrical pores with diameters that can be tailored within a range of 10– 80  nm.^{[20, 21]}

The porosity ϕ is introduced to denote the volume ratio of pores in mesoporous alumina, which is given by where r is the pore radius, r=d/2, with d being the mesopore diameter, and t the half the wall thickness. As the diameter of the pore is much smaller than λ _T, the electromagnetic field across the pore is dominated by the near-field radiation. λ _T is given by λ _T ≈ ħ c/k_BT, and is about several micrometers at room temperature.

Since the mesoporous alumina has a periodical structure, a structure element containing a cylindrical pore can be suggested as shown in Fig.  1(b). The pore is characterized by the free-space permittivity ε ₀, and bounded by alumina with relative permittivity ε ₁. The temperature distribution can be expressed as T(ρ , θ )= 1/2[(T₁− T₂)ρ cos θ /r + T₁ + T₂].^[23] It drives radiative heat transfer across the pore, where T₁ is the high temperature at θ = 0° , T₂ is the low temperature at θ = 180° on the pore boundary. As illustrated in Fig.  1(b), heat radiated from a source located at (ρ , θ ) transfers to the point (ρ ′ , θ ′ ) by traveling a distance L= [(ρ ² + ρ ′ ²− 2ρ ρ ′ cos(θ − θ ′ )]^1/2.

Fig.  1.

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	Fig.  1. (a) Structure of mesoporous alumina. where red spheres represent oxygen atoms, yellow spheres denote aluminium atoms). (b) Simplified a pore for near-field radiation calculation.

Unlike the far-field radiation (macroscale thermal radiation) transferred by the propagating of the transmissible electromagnetic waves, the near-field radiation is transferred by photon tunneling of the evanescent waves.^{[24, 25]} Now considering a monochromatic electromagnetic wave in the vacuum region, the complex wave vector in a vacuum can be expressed as κ ₀ = (β , γ ₀), which can be seen in Fig.  2, and κ ₀ = ω /c, where c is the speed of light in vacuum, β is the parallel component of the wave vector that is parallel to the plane, γ ₀ is the perpendicular component of the wave vector, ω is the angular frequency of the electromagnetic wave.

Fig.  2.

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	Fig.  2. Complex wave vector of a monochromatic electromagnetic wave in the vacuum.

Assuming that thermal and quantum fluctuations of the current density j inside the body create the fluctuating electromagnetic field, it is known that the time-averaged〈 j 〉 = 0 but 〈 j · j 〉 ≠ 0. We can describe the fluctuating electromagnetic field by the Maxwell equations with the fluctuating current density. According to the electromagnetic wave theory, for a monochromatic wave in a dielectric, isotropic, nonmagnetic body, the Maxwell equations corresponding to radiative heat transfer can be shown as

where E is the electric field, H the magnetic field, ε ₀ the vacuum permittivity, μ ₀ the magnetic permeability of vacuum, ε ₁ the relative permittivity(dielectric constant) of alumina, and j the fluctuating current density. Considering boundary conditions, the fluctuating electromagnetic field can be obtained, then the frequency-dependent spectral Poynting vector, i.e., spectral heat flow across the pore, is given by

where * denotes the complex conjugate.

Using the fluctuating current density as the source, from the Maxwell equations the electric field can be given by

where κ ₁ is the wave vector in alumina, , c₁ is the speed of light in alumina.

When we calculate the radiative heat transfer across a cylindrical pore, the Fourier transform of the three-dimensional expression from z to κ _1Z space is used, and set κ _1Z = 0. The induced electric and magnetic fields caused by the fluctuating current density can be expressed as a frequency-dependent volumetric integral, according to the dyadic Green's function G(ρ , ρ ′ , ω ):

where G(ρ , ρ ′ , ω ) is essentially a spatial transfer function between a fluctuating current source j at a location ρ ′ and the electric field E at another location ρ . The integration is over the region V that contains the source of fluctuating current density.

The radiative heat transfer between the left side and the right side of a curved interface is calculated by employing the dyadic Green’ s function characterizing the electric field, G(ρ , ρ ′ , ω ), which is given as the solution to a inhomogeneous cylindrical Helmholtz equation. The dyadic Green's function must satisfy the boundary condition at the vacuum-silica interface (ρ = r), which can be given by ∇ × G(ρ , ρ ′ , ω ) = iκ ₀G(ρ , ρ ′ , ω ). We can obtain the electric Green’ s functions, ^[23] which can be shown as

where the reflection coefficient can be shown as

where J_n and are Bessel functions of the first and third kind respectively, and The first term on the right-hand side of Eq.  (5) is the field incident on the curved interface, while the second term is the field scattered away. The Green function in Eq.  (5) describes the field transmitted from the alumina, through the boundary and into the vacuum.

In this paper, the pore diameter in our model is greater than 1  nm since the dielectric function is not local nor dependent on wave vector^[15] when the pore diameter is less than 1  nm. Therefore, we can assume that the dielectric function is a scalar (isotropic medium) and independent of the wave vector. The dielectric function can be written as ε ₁(ω ), which can be calculated by the data in handbook of optical constants.^[26]

The fluctuating current density within the body, as the source, can be described by fluctuation-dissipation theorem (FDT). It is related to the alumina equilibrium temperature T and dielectric function ε ₁(ω ). The random thermal fluctuations of charges create a space- and time-dependent fluctuating electric current density inside the silica whose time average is zero. The time-dependent current density can be decomposed into j(ρ , ω ) in the frequency-dependent form by using the Fourier transform. The ensemble average of the spatial correlation function of current density^[27] can be obtained by

where j(ρ , ω ) is the component of fluctuating current density at point ρ , with ω being the angular frequency; δ is the Dirac delta function; 〈 〉 denotes the ensemble averaging, which is an average concept in quantum mechanics.

According to Eqs.  (2), (4), (5), and (7), we can obtain the Poynting vector, i.e., radiative heat flow across the pore, which is shown (in units of W· md^{− 1}) as follows:

where the last term on the right-hand side refers to the net radiative heat flux from to the location ρ (ρ = r, θ = π ). The radiative heat flux is given by q = S/A (in units of W· m^{− 2}), with A being the near-field radiation area.

Finally, equivalent thermal conductivity of radiative heat transfer (in units of W· m^{− 1}· K^{− 1}) can be obtained as follows:

where Δ T = T₁− T₂.

The combined thermal conductivity of mesoporous alumina is a combination of the thermal conductivity of mesoporous alumina (substrate thermal conductivity), k_S, the thermal conductivity of void space (air), k_g, and the equivalent thermal conductivity of radiative heat transfer across mesopore, k_r.

The thermal conductivity of mesoporous alumina, k_S, is lower than that of normal bulk alumina (k₀ = 33  W · m^{− 1}· K^{− 1})^[28] since mesoporous alumina is not a crystal but an amorphous material. The kinetic theory is used to calculate the substrate thermal conductivity in the X– Y plane and along the Z direction, and it is shown as follows:^[29]

where C_V is the heat capacity, τ _XY/Z is the combined relaxation time, and v is the average sound velocity. So the reduced substrate thermal conductivity scaled by that of the corresponding bulk can be written as

The superscript “ * ” above means a dimensionless quantity scaled by the corresponding quantity of the bulk material. The scaled specific heat capacity of a nanoporous material is a function of the pore radius r and the porosity ϕ , ^[21]

The values of equation parameters can be obtained from Ref. [20].

The combined phonon relaxation time is obtained from the Matthiessen rule as^[30]

On the right-hand side of the above equation, the first term is the reciprocal of the relaxation time corresponding to the scattering in the normal bulk alumina, and l₀ is the phonon mean free path of the normal bulk alumina; the second term is the reciprocal of the relaxation time corresponding to the defect scattering, and l₁ is a fitting parameter which shows the glass defect levels; in the third term, τ _{B, XY/Z} is the relaxation time corresponding to the boundary scattering. The boundary relaxation rate along the Z direction^[31] can be expressed as

The boundary relaxation rate in the X– Y plane can be expressed as

According to Eqs.  (11)– (15), we obtain the scaled substrate thermal conductivity as

The substrate thermal conductivity can be written as k_S = 2/3k_{s, xy} + 1/3k_{S, Z}.

Zeng et al.’ s model^[32] is used to describe the thermal conductivity of confined air in mesopore, k_g, as

where the parameter γ = c_P/c_V, N_g0 = n_g/V ϕ , A_s is the surface area of the shell, V_s the apparent volume of the material, m_g the mass of an air molecule, n_g the number density of the air molecules, and d_g the diameter of an air molecule. An air molecule has a mass of 4.648× 10^{− 26}  kg and a diameter of 3.53× 10^{− 10}  m.^[33]

The porosity weighted dilute medium (PWDM) model^[22] is used to obtain the combined thermal conductivity of mesoporous alumina by taking into account the thermal conduvtivities of alumina, confined air and near-field radiation. The PWDM model represents thermal conductivity variations of the dilute particle, fluid or other porous materials, through employing a combination of the dilute particle and dilute fluid models, so it is more accurate than the porosity weighted simple medium (PWSM) model, ^[22] which is a combination of the parallel and serial models. Accordingly, the combined thermal conductivity of mesoporous alumina can be expressed as

where x is the semi-empirical fitting parameter, which ranges from zero to infinity and accounts for the effects of pore shape, pore size, and other parameters; k_f = k_r+k_g.

Near-field radiation has great influence on mesoporous material thermal properties. We focus on the near-field radiative heat transfer across a mesopore and its contribution to the combined thermal conductivity of mesoporous alumina. The main findings and conclusions are summarized as follows.