Influence of roughness on near-field heat transfer between two plates

The surface roughness correction to the near-field heat transfer between two rough bulk materials is discussed by using second-order perturbation theory. The results allow for estimating the impact of surface roughness to the heat transfer in recent experiments between two plates and between a microsphere and a plate (using the Derjaguin approximation). Furthermore, we show that the proximity approximation for describing rough surfaces is valid for distances much smaller than the correlation length of the surface roughness even if the heat transfer is dominated by the coupling of surface modes.

Figure 1

Sketch of the situation considered here.

Figure 2

(Color online) Plot of (a) $S^{\left(0\right)}$ over distance for two SiC plates and two Au plates normalized to the black body value $S_{\text{BB}}$ with (b) the corresponding roughness correction $\Delta S$.

Figure 3

(Color online) Plot of the specular and diffuse part of the roughness correction $\left|\Delta ⟨S⟩\right|$ over distance for SiC.

Figure 4

Sketch of the diffuse scattering in terms of transmission coefficients. Here $Q_p^1\propto T_p^{\left(0\right)}\left(\kappa \right)$ and $Q_p^{1\prime }\propto T_p^{\left(0\right)}\left({\kappa }^{\prime }\right)$.

Figure 5

(Color online) Plot of (a) $T_p^{\left(0\right)}$ as defined in Eq. (15) and (b) $T_{p,\text{spec}}^{\left(2\right)}/T_p^{\left(0\right)}$ in the propagating regime using $d=5\mu \text{m}$ for frequencies ranging from $\omega ={\omega }_{\text{l}}=1.83\times {10}^{14}$ to $7\times {10}^{14}{\text{s}}^{-1}$.

Figure 6

(Color online) Plot of (a) $T_p^{\left(0\right)}$, (b) $T_{p,\text{spec}}^{\left(2\right)}/T_p^{\left(0\right)}$, and (c) $T_{p,\text{diff}}^{\left(2\right)}/T_p^{\left(0\right)}$ using $d=500\text{nm}$.

Figure 7

(Color online) Plot of $⟨S_{\omega }^{\left(2\right)}⟩/S_{\omega }^{\left(0\right)}$ in a small frequency range around the surface phonon frequency of SiC $\left({\omega }_{\text{SPhP}}=1.787\times {10}^{14}{\text{s}}^{-1}\right)$ using $d=5\mu \text{m}$. We have plotted the specular (solid line/red line) and the diffuse (dashed line/green line) contributions (note the different scales at the right and left-hand side) and $S_{\omega }^{\left(0\right)}$ (dotted line/blue line).

Figure 8

(Color online) As Fig. 7 using $d=500\text{nm}$.

Figure 9

Sketch of the proximity approximation.

Figure 10

(Color online) Plot of $S\left(d\right)/S_{\text{PA}}\left(d\right)$ for SiC over $a$ keeping the rms constant at $\delta =5\text{nm}$ for $d=200$, 250, 500, and 750 nm.

Figure 11

(Color online) As Fig. 2b but with the PA for SiC and Au calculated from $S^{\left(0\right)}$ with Eq. (23).