Near-field radiative heat transfer between arbitrarily shaped objects and a surface

A fluctuational electrodynamics-based formalism for calculating near-field radiative heat transfer between objects of arbitrary size and shape and an infinite surface is presented. The surface interactions are treated analytically via Sommerfeld's theory of electric dipole radiation above an infinite plane. The volume integral equation for the electric field is discretized using the thermal discrete dipole approximation (T-DDA). The framework is verified against exact results in the sphere-surface configuration and is applied to analyze near-field radiative heat transfer between a complex-shaped probe and an infinite plane, both made of silica. It is found that, when the probe tip size is approximately equal to or smaller than the gap $d$ separating the probe and the surface, coupled localized surface phonon (LSPh)-surface phonon-polariton (SPhP) mediated heat transfer occurs. In this regime, the net spectral heat rate exhibits four resonant modes due to LSPhs along the minor axis of the probe, while the net total heat rate in the near field follows a $d^{-0.3}$ power law. Conversely, when the probe tip size is much larger than the separation gap $d$, heat transfer is mediated by SPhPs, resulting in two resonant modes in the net spectral heat rate, corresponding to those of a single emitting silica surface, while the net total heat rate approaches a $d^{-2}$ power law. It is also demonstrated that a complex-shaped probe can be approximated by a prolate spheroidal electric dipole when the thermal wavelength is larger than the major axis of the spheroidal dipole and when the separation gap $d$ is much larger than the radius of curvature of the dipole tip facing the surface.

Figure 1

Schematic representation of the problem under consideration. Objects (medium 2) are submerged in vacuum (medium 0) above an infinite surface (medium 1). The incident electric field ${\mathbf{E}}^{\mathrm{inc}}$ accounts for illumination by external sources, while the surface field ${\mathbf{E}}^{\mathrm{sur}}$ is the electric field due to thermal emission by the surface.

Figure 2

Dyadic Green's function (DGF) relating the electric field at point r to a source located at point ${\mathbf{r}}^{\prime }$ in the presence of a surface.

Figure 3

Dielectric function of silica obtained from Ref. [53].

Figure 4

Net spectral heat rate between a sphere and a surface for separation gaps $d$ of 100 nm and 100 μm obtained with the T-DDA and the exact solution [10]. The sphere is at a temperature $T_2=400\mathrm{K}$, while the surface is at $T_1=300\mathrm{K}$. The sphere and the surface are made of silica, and the sphere diameter is 1.6 μm.

Figure 5

Radiative heat transfer between a probe and a surface separated by a gap of thickness $d$. The probe is modeled as an assembly of a rectangular cuboid, a conical frustum, and a cylinder.

Figure 6

Net spectral heat rate between (a) a probe and a surface, and (b) a sphere and a surface for separation gaps $d$ of 10 nm, 100 nm, and 100 μm. The sphere and the probe are at temperature $T_2=400\mathrm{K}$, while the surface is at temperature $T_1=300\mathrm{K}$. The sphere diameter is 1.6 μm, and the probe dimensions are shown in Fig. 5. The sphere, the probe, and the surface are made of silica.

Figure 7

Spectral distribution of energy density at distances $d$ of 10 nm, 100 nm, and 100 μm above a silica surface at a temperature $T_1=300\mathrm{K}$.

Figure 8

Net total heat rate in the near field as a function of the separation gap $d$ for the probe-surface and sphere-surface configurations. The sphere and the probe are at $T_2=400\mathrm{K}$, while the surface is at $T_1=300\mathrm{K}$. The sphere diameter is 1.6 μm, and the probe dimensions are shown in Fig. 5. The sphere, the probe, and the surface are made of silica.

Figure 9

Spatial distribution of normalized volumetric heat rate within the probe at the low-frequency resonance for gap sizes $d$ of 10 nm, 100 nm, and 100 μm. The probe and the surface are made of silica.

Figure 10

Net spectral heat rate between a probe and a surface for gap sizes $d$ of 10 and 100 nm. Results are compared against the spheroidal electric dipole model. The probe is at $T_2=400\mathrm{K}$, while the surface is at $T_1=300\mathrm{K}$. The probe and the surface are made of silica.

Figure 11

Net total heat rate between a probe and a surface as a function of the separation gap $d$. Results are compared against the spheroidal electric dipole model. The probe is at $T_2=400\mathrm{K}$, while the surface is at $T_1=300\mathrm{K}$. The probe and the surface are made of silica.