Many-body heat radiation and heat transfer in the presence of a nonabsorbing background medium

Heat radiation and near-field radiative heat transfer can be strongly manipulated by adjusting geometrical shapes, optical properties, or the relative positions of the objects involved. Typically, these objects are considered as embedded in vacuum. By applying the methods of fluctuational electrodynamics, we derive general closed-form expressions for heat radiation and heat transfer in a system of $N$ arbitrary objects embedded in a passive nonabsorbing background medium. Taking into account the principle of reciprocity, we explicitly prove the symmetry and positivity of transfer in any such system. Regarding applications, we find that the heat radiation of a sphere as well as the heat transfer between two parallel plates is strongly enhanced by the presence of a background medium. Regarding near- and far-field transfer through a gas like air, we show that a microscopic model (based on gas particles) and a macroscopic model (using a dielectric contrast) yield identical results. We also compare the radiative transfer through a medium like air and the energy transfer found from kinetic gas theory.

Figure 1

The system of $N$ arbitrary objects embedded in a passive nonabsorbing background medium in thermal nonequilibrium as considered in this paper. The objects are subject to thermal charge and current fluctuations (indicated by the white arrows). The resulting electromagnetic field fluctuations (mimicked by the red arrows) give rise to typical nonequilibrium phenomena, such as heat radiation and heat transfer between objects.

Figure 2

Illustration of the quantity $H_{\alpha }^{\left(\beta \right)}$ in a many-body system embedded in a passive nonabsorbing background medium. The heat radiation emitted by object $\alpha$ (red) at temperature $T_{\alpha }$ is partially absorbed by object $\beta$ (blue) at temperature $T_{\beta }$. The residual objects are marked by a gray color.

Figure 3

The “self”-emission $H_{\beta }^{\left(\beta \right)}$ in a many-body system embedded in a passive nonabsorbing background medium. The heat radiation emitted by object $\beta$ (red) at temperature $T_{\beta }$ is backscattered at the residual objects (gray) and partially reabsorbed.

Figure 4

Main graph: heat radiation of a gold sphere at $T=300\text{K}$ in a homogeneous nonabsorbing background medium with dielectric constant ${\varepsilon }_b$ as given, as a function of radius $R$, normalized by the result in vacuum $\left({\varepsilon }_b=1\right)$. The inset graph shows the corresponding result in vacuum, normalized by the Stefan-Boltzmann result (this curve is identical to the one in Fig. 2 of Ref. [14]).

Figure 5

Heat radiation of a gold sphere at $T=300\text{K}$ in a homogeneous nonabsorbing background medium as a function of dielectric constant ${\varepsilon }_b$, normalized by the result in vacuum (see inset graph in Fig. 4).

Figure 6

Main graph: heat radiation of a ${\mathrm{SiO}}_2$ sphere at $T=300\text{K}$ in a homogeneous nonabsorbing background medium with dielectric constant ${\varepsilon }_b$ between 1.2 and 2 as a function of radius $R$, normalized by the result in vacuum. The inset graph shows the corresponding result in vacuum, normalized by the Stefan-Boltzmann result (this curve is identical to the one in Fig. 2 of Ref. [14]).

Figure 7

Heat radiation of a ${\mathrm{SiO}}_2$ sphere at $T=300\text{K}$ in a homogeneous nonabsorbing background medium as a function of dielectric constant ${\varepsilon }_b$, normalized by the result in vacuum (see inset graph in Fig. 6).

Figure 8

Two semi-infinite half-spaces held at different temperatures $T_1>T_2$ and separated by a gap of length $d$. The gap is filled with a nonabsorbing homogeneous background medium.

Figure 9

Heat transfer between two SiC plates (upper curve) and two gold plates (lower curves) embedded in vacuum (i.e., ${\varepsilon }_b=1)$ as a function of the plate distance $d$. The plates are held at the homogeneous temperatures $T_1=301\text{K}$ and $T_2=300\text{K}$. On the submicron scale $d\ll {\lambda }_{T_1}\approx 8\mu \mathrm{m}$, the transfer is strongly enhanced because of photon tunneling processes. The dashed line shows the approximation for two SiC plates in the small-distance limit given by Eq. (60).

Figure 10

Heat transfer between two SiC plates at the homogeneous temperatures $T_1=301\text{K}$ and $T_2=300\text{K}$ embedded in a nonabsorbing medium as a function of the plate distance $d$. The transfer is normalized by the result in vacuum (see upper curve in Fig. 9).

Figure 11

Heat transfer between two gold plates at the homogeneous temperatures $T_1=301\text{K}$ and $T_2=300\text{K}$ embedded in a nonabsorbing medium as a function of the plate distance $d$. The transfer is normalized by the result in vacuum (see lower curve in Fig. 9).

Figure 12

Enhancement factor of the heat transfer between two SiC and two gold plates, respectively, in a background medium compared to the vacuum case. Circles symbolize the transfer in the near field $\left(d=1\text{nm}\right)$, while squares show plate separations $d=10\mu \mathrm{m}$ (far field). The temperatures of the plates were chosen identically to Figs. 10 and 11.

Figure 13

The arrangement of two semi-infinite half-spaces and an atom in vacuum. The half-spaces are kept at different temperatures $T_1>T_2$ and separated by a gap of length $d$. The absorptivity of the atom (or nanoparticle) is assumed to be negligible, such that it only scatters the (near and far) fields radiated by the plates.

Figure 14

Correction term $\mathrm{\Delta }{H}^{1\to 2}$ to the heat transfer between two SiC plates in vacuum at distance $d=10\text{nm}$ due to the presence of a polarizable atom over the atom position $d_1$. The plates are held at the homogeneous temperatures $T_1=301\text{K}$ and $T_2=300\text{K}$. The correction is normalized by the polarizability $\alpha$ of the atom, which is assumed to be a nondispersive real quantity.

Figure 15

Correction term $\mathrm{\Delta }{H}^{1\to 2}$ to the heat transfer between two SiC plates in vacuum at distance $d=10\mu \text{m}$ due to the presence of a polarizable atom over the atom position $d_1$. The other parameters are chosen identically to Fig. 14.

Figure 16

Comparison between the microscopic and the macroscopic model for the change in the heat transfer due to the presence of air between two SiC half-spaces at temperatures $T_1=301\text{K}$ and $T_2=300\text{K}$. The completely different approaches yield the same result in the dilute limit.

Figure 17

Energy fluxes in the system of two plates embedded in air. The radiative component $H_{\mathrm{air}}^{1\to 2}$ surpasses the kinetic part $H_{\mathrm{kin}}$ both in the near and in the far field. Increasing the temperature of the plates shifts the point of intersection of the two contributions to larger plate separations $d$ in the near field and to smaller plate separations in the far field, respectively.