Fluctuating-surface-current formulation of radiative heat transfer: Theory and applications

We describe a fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to nonequilibrium fluctuations that involve scattering matrices—relating “incoming” and “outgoing” waves from each body—our approach is formulated in terms of “unknown” surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semianalytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g., Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g., interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, such as the heat transfer between finite slabs, cylinders, and cones.

Figure 1

Schematic depicting two disconnected bodies described by surfaces $\partial V^1$ and $\partial V^2$ and held at temperature $T^1$ and $T^2$, respectively. Surface currents ${\xi }^1$ and ${\xi }^2$ laying on the surfaces of the bodies give rise to scattered fields ${\varphi }^{1-}$ and ${\varphi }^{2-}$, respectively, in the interior of the bodies, and scattered field ${\varphi }^{0-}$ in the intervening medium 0.

Figure 2

Schematic depicting three disconnected bodies described by surfaces $\partial V^1$, $\partial V^2$, and $\partial V^3$, and held at temperature $T^1$, $T^2$, and $T^3$, respectively. Surface currents ${\xi }^1$, ${\xi }^2$, and ${\xi }^3$, laying on the surfaces of the bodies give rise to scattered fields ${\varphi }^{1-}$, ${\varphi }^{2-}$, and ${\varphi }^{3-}$, respectively, in the interior of the bodies, and scattered field ${\varphi }^{0-}$ in the intervening medium 0.

Figure 3

Similar three-body geometry as that depicted in Fig. 2 but with body 3 now embedded in the interior of body 2. Here, the scattered field ${\varphi }^{2-}$ includes contributions from both ${\xi }^2$ and ${\xi }^3$.

Figure 4

Ratio $\mathcal{H}=H/\sigma T^{4}A$ of the heat-transfer rate $H$ between gold spheres of radii $R=0.2\mu$m and the Stefan-Boltzmann law $\sigma T^{4}A$, where $A=4\pi R^2$ is the surface area of the spheres, with one sphere held at $T=300$ K and the other held at zero temperature, as a function of their surface-surface separation. (Inset) Flux spectra $\Phi \left(\omega \right)$ per unit area $A$ (units of $\mu {\mathrm{m}}^2$) of the two spheres at $d=R$ (green circles) and of an isolated sphere (blue circles).

Figure 5

Ratio $\mathcal{H}=H/\sigma T^{4}A$ of the heat-transfer rate $H$ from a finite, gold circular plate of lateral size $2R$ and thickness $L=0.2\mu$m held at $T=300$ K, to an identical plate held at zero temperature, and the SB law $\sigma T^{4}A$ (where $A=\pi R^2$ is the “interaction” surface area of the plates), as a function of their surface-surface separation $d$. $\mathcal{H}$ is plotted for multiple aspect ratios $2R/L$ (circles). The solid black line corresponds to the heat-transfer ratio ${\mathcal{H}}_{\infty }={\mathcal{H}}_{\infty }(R\to \infty )$ obtained upon taking the limit $R\to \infty$, which is computed via the semianalytical formula in Ref. 109. (Inset) Heat-transfer rate between two plates at a fixed separation $d=0.1L$ (solid circles) and heat radiation of an isolated plate (open circles) or sphere (thick solid line) as a function of lateral size or diameter.

Figure 6

Flux spectrum $\Phi \left(\omega \right)$ of an isolated gold cylinder of length $L$ and radius $R=0.2\mu$m normalized by its corresponding surface area $A$ and plotted for multiple aspect ratios $L/R$ (solid circles). The solid line shows $\Phi$ in the limit $L\to \infty$ of a semi-infinite cylinder, as computed by the semianalytical formula of Ref. 34. (Inset) Heat-transfer ratio $\mathcal{H}$ of heat transfer $H$ from a room-temperature cylinder of aspect ratio $L/R=5$ to an identical cylinder at zero temperature, and the SB law $\sigma T^{4}A$, with $A=2\pi R(R+L)$ denoting the total surface area of each cylinder, as a function of their surface-surface separation $d$. $H$ is plotted for both parallel ($\theta =0$) and crossed ($\theta ={90}^{\circ }$) cylinder configurations, with the shaded region corresponding to intermediate $\theta$.

Figure 7

Heat-transfer rate $H$ from a room-temperature gold cone of base radius $R=1\mu$m and length $L=2R$, to either an identical cone (green circles) or a gold plate of radius $R$ and thickness $h=0.2\mu$m, held at zero temperature, as a function of their surface-surface separation $d$. $H$ is normalized by the Stefan-Boltzmann law $\sigma T^{4}A$, where $A$ is the surface area of the cone. (Inset) Flux spectrum $\Phi \left(\omega \right)$ of the cone-plate configuration at a single separation $d=0.2L$, normalized by the area of the cone. The two surface-contour plots show the distribution of flux pattern on the surfaces of the bodies at two wavelengths, $\lambda \approx 30L$ and $\approx$$2.2L$, where white/black denotes the maximum/minimum flux at the corresponding wavelength.