The near field radiative heat transfer (NFHT) is calculated between a SiC half-space at temperature *T*_{2}=1 *K* and the composite slab at a temperature *T*_{1}=300 *K*, both slabs separated by a distance *d*. In the case of the composite slab, the nanospheres are randomly distributed and the effective dielectric function is isotropic. Using the known results of fluctuating electrodynamics [26], [27], the heat flux *Q* is calculated as

$$Q=\frac{1}{4{\pi}^{2}}{\displaystyle \int \left[\left[\theta \mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}{T}_{1}\mathrm{)}-\theta \mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}{T}_{2}\mathrm{)}\right]{\displaystyle \sum _{p\mathrm{,}s}{\displaystyle \int q{\tau}_{p\mathrm{,}s}\mathrm{(}\omega \mathrm{,}q\mathrm{,}d\mathrm{)}dq}}\right]d\omega}\mathrm{,}$$(6)

where *θ*(*ω*, *T*)=*ħω*/(*exp*(*ħω*/*k*_{B}*T*)−1) is Planck’s distribution function, *q* is the parallel component to the slabs of the wave vector, and the perpendicular component is ${k}_{0}=\sqrt{{\omega}^{2}\mathrm{/}{c}^{2}-{q}^{2}}.$ For either *p* or *s* polarised waves, the transmission coefficient is

$${\tau}_{p\mathrm{,}s}\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}q\mathrm{,}\text{\hspace{0.17em}}L\mathrm{)}=\mathrm{(}\begin{array}{cc}\mathrm{(}1-\mathrm{|}{r}_{p\mathrm{,}s}^{1}{\mathrm{|}}^{2}\mathrm{)}\mathrm{(}1-\mathrm{|}{r}_{p\mathrm{,}s}^{2}{\mathrm{|}}^{2}\mathrm{)}/|{D}_{p\mathrm{,}s}^{12}{\mathrm{|}}^{2}\mathrm{,}& \text{if\hspace{0.17em}}q<\omega \text{/}c\\ 4Im\mathrm{(}{r}_{p\mathrm{,}s}^{1}\mathrm{)}Im\mathrm{(}{r}_{p\mathrm{,}s}^{2}\mathrm{)}{e}^{\mathrm{(}-2{k}_{0}d\mathrm{)}}/|{D}_{p\mathrm{,}s}^{12}{\mathrm{|}}^{2}\mathrm{.}& \text{if\hspace{0.17em}}q>\omega \text{/}c\end{array}$$(7)

The terms ${r}_{p\mathrm{,}s}^{1}$ and ${r}_{p\mathrm{,}s}^{2}$ are the Fresnel coefficients for the slabs at temperature *T*_{1} and *T*_{2}, respectively. The Fabry-Perot type term in the denominator is ${D}_{p\mathrm{,}s}^{12}=\mathrm{|}1\mathrm{-}{r}_{p\mathrm{,}s}^{1}{r}_{p\mathrm{,}s}^{2}\mathrm{exp}\mathrm{(}2i{k}_{0}d\mathrm{)}|\mathrm{.}$

Setting the temperatures at *T*_{1}=300 *K* and *T*_{2}=1 *K*, we calculate the heat flux as a function of the separation *d* for the different effective medium models at various filling fractions of the nanospheres. In Figure 1b, we show the configuration of the system. In Figure 5, we present the values of the heat flux, normalised to the far field value ${Q}_{bb}=\sigma \mathrm{(}{T}_{1}^{4}-{T}_{2}^{4}\mathrm{)},$ where *σ*=5.670×10^{−8} Wm^{−2} K^{−4}. As a reference, the value of *Q* for two homogeneous SiC slabs is also plotted. For a filling fraction of *f*=0.010=1%, we see that there is no significant difference between the different effective medium models, except for Looyengas that shows a smaller value of *Q* for all separations. As the filling fraction increases to *f*=0.1 and *f*=0.15, we observe differences on the predicted value of *Q* depending on the model. Looyenga’s approximation gives a lower bound for the total near field radiative heat transfer for all values of the filling fractions. At a filling fraction of *f*=0.33, the Bruggeman model predicts a percolation transition, that corresponds to the close packing of the spheres. At this filling fraction, we see the largest differences in the predicted values of *Q*.

Figure 5: Near field radiative heat flux normalised to the black body ideal case ${Q}_{bb}=\sigma \mathrm{(}{T}_{1}^{4}-{T}_{2}^{4}\mathrm{)}$ as a function of slab separations. The panels correspond to different values of the filling fraction, as indicated (a) *f*=0.01, (b) *f*=0.1, (c) *f*=0.15 and (d) *f*=0.33. As a reference, the solid line is the NFHT between two homogeneous SiC slabs.

For a system with a small filling fraction, Maxwell-Garnett does not match the experimental measurements of the optical properties for the systems studied by Grundquist and Hunderi. Neither does the Bruggeman approximation. Grundquist introduced an ad-hoc correction [13], [14] to the Maxwell-Garnett model. The effective dielectric function in this case is

$${\tilde{\u03f5}}_{G}={\u03f5}_{h}{\mathrm{(}\frac{4+f\alpha}{4-f\alpha}\mathrm{)}}^{2}\mathrm{,}$$(8)

where

$$\alpha =3\frac{{\u03f5}_{i}-{\u03f5}_{h}}{{\u03f5}_{i}+2{\u03f5}_{h}}\mathrm{.}$$(9)

For comparison, we can write Maxwell-Garnett equation (1) as

$${\tilde{\u03f5}}_{MG}={\u03f5}_{h}\frac{1+2f\alpha}{1-f\alpha}\mathrm{.}$$(10)

For two different filling fractions, we show in Figure 6 the total heat flow obtained using Maxwell-Garnett (8) of the Grundquist-Hunderi (10) EMAs. At a filling fraction of *f*=0.1, the value of *Q* differs for both models for all separations considered. However, for *f*=0.15, the difference becomes smaller. This is because the larger the amount of Au in the sample, the effective dielectric function becomes more metallic and the dielectric function of Au begins to be more prevalent in the effective response.

Figure 6: Comparison of the near field radiative heat flux normalised to the black body ideal case ${Q}_{bb}=\sigma \mathrm{(}{T}_{1}^{4}-{T}_{2}^{4}\mathrm{)}$ as a function of slab separations for the Maxwell-Garnett and the Grundquist-Hunderi models for filling fractions *f*=0.1 (a) and *f*=0.15 (b). Although the two models are very similar, there are differences in the predicted values of *Q*.

## Comments (0)