Strongly coupled near-field radiative and conductive heat transfer between planar bodies

We study the interplay of conductive and radiative heat transfer (RHT) in planar geometries and predict that temperature gradients induced by radiation can play a significant role on the behavior of RHT with respect to gap sizes, depending largely on geometric and material parameters and not so crucially on operating temperatures. Our findings exploit rigorous calculations based on a closed-form expression for the heat flux between two plates separated by vacuum gaps $d$ and subject to arbitrary temperature profiles, along with an approximate but accurate analytical treatment of coupled conduction-radiation in this geometry. We find that these effects can be prominent in typical materials (e.g., silica and sapphire) at separations of tens of nanometers, and can play an even larger role in metal oxides, which exhibit moderate conductivities and enhanced radiative properties. Broadly speaking, these predictions suggest that the impact of RHT on thermal conduction, and vice versa, could manifest itself as a limit on the possible magnitude of RHT at the nanoscale, which asymptotes to a constant (the conductive transfer rate when the gap is closed) instead of diverging at short separations.

Figure 1

The left inset shows a schematic of two parallel slabs separated by a distance $d$ along the $z$ direction ($z=0$ denotes the middle of the gap). The two slabs exhibit a temperature profile $T\left(z\right)$: the temperature is constant and has values $T_L$ and $T_R$ in the regions $z\le -d/2-t_a$ and $z\ge d/2+t_b$, respectively, and varies linearly in the regions $-d/2-t_a<z<-d/2$ and $d/2<z<d/2+t_b$, with $T_a$ and $T_b$ denoting the temperatures at the slab-vacuum interfaces. The main part of the figure shows the temperature profile along the temperature-varying region of the hot slab in a configuration involving two silica slabs with $t_a=t_b=100\mu \mathrm{m}$ held at external temperatures $T_L=600\mathrm{K}$ and $T_R=300\mathrm{K}$ at the outer ends, and separated by vacuum gaps $d$. The points $z=0,100\mu \mathrm{m}$ represent the boundary of the thermostat and the interface with the vacuum gap. The three lines correspond to $d=10\mathrm{nm}$ (black), 20 nm (red), and 50 nm (blue). The right inset shows the $z$-dependent radiative flux rate $\phi \left(z\right)$ (see text) along slab $a$ at $d=100\mathrm{nm}$.

Figure 2

Total flux $\phi$ and temperature difference $T_a-T_b$ (inset) as a function of distance $d$ between two silica slabs (shown schematically in the left inset of Fig. 1) that are being held at $T_L=600\mathrm{K}$ and $T_R=300\mathrm{K}$. The various solid lines correspond to different temperature-varying regions $t$ (from top to bottom): 100 nm (black), $1\mu \mathrm{m}$ (red), $10\mu \mathrm{m}$ (brown), $100\mu \mathrm{m}$ (blue), and $500\mu \mathrm{m}$ (green). The orange dashed line shows $\phi$ in the absence of temperature gradients.

Figure 3

Typical distance scale $\tilde{d}$ relevant to conduction-radiation problems (see text) as a function of the material-dependent ratio $h_0/\kappa$ for two slabs with $t_a=t_b=100\mu \mathrm{m}$. Solid circles denote corresponding values for specific material choices (abbreviations), with AZO[1.2] and AZO[0.05] denoting aluminum zinc oxides of different conductivities $\kappa =1.2\mathrm{W}/\mathrm{m}\mathrm{K}$ [52] and $\kappa =0.05\mathrm{W}/\mathrm{m}\mathrm{K}$ [15], respectively. The inset shows the dependence of the radiative-heat transfer coefficient $h_0$ on the external temperature gradient $\mathrm{\Delta }T=T_L-300\mathrm{K}$, for three cases, AZO (red), silica (blue), and SiC (black), identified by their decreasing values.