Limitations of kinetic theory to describe near-field heat exchanges in many-body systems

We investigate the radiative heat transfer along a chain of nanoparticles using both a purely kinetic approach based on the solution of a Boltzmann transport equation and an exact method (Landauer's approach) based on fluctuational electrodynamics for temperatures close to room temperature. We show that the kinetic theory generally fails to predict properly the heat flux transported along the chain at both close (near-field regime) and large-separation (far-field regime) distances. We report a deviation of a factor of 2 between the heat fluxes predicted by the two approaches in the diffusive regime of heat transport, and we show that this difference becomes even greater than two orders of magnitude in the ballistic regime. In the case of metallic nanoparticles, we show that the kinetic approach completely fails in describing the magnetic contribution and gives dramatically wrong results for the electric one.

Figure 1

Sketch of the considered configuration: a set of $N$ spherical nanoparticles aligned along the $z$ axis. The particles all have radius $a$, while $d$ is the center-to-center distance between neighboring particles.

Figure 2

Dispersion relation for a chain of hBN nanoparticles with radius $a=25\mathrm{nm}$ and distance $d=4a=100\mathrm{nm}$ for transverse (black curves) and longitudinal (blue curves) polarizations. The lines correspond to the search for complex frequency $\omega$ (the plot shows the real part of $\omega$) using real $k$, while the symbols correspond to the quasistatic limit.

Figure 3

Propagation length for transverse (black) and longitudinal (blue) polarizations for a chain of hBN nanoparticles with radius $a=25$ nm. Dashed lines show the results for $d=3a=75\mathrm{nm}$, whereas solid lines correspond to $d=7a=175\mathrm{nm}$.

Figure 4

Normalized propagation length at the resonance frequency ${\omega }_0$ as a function of the normalized distance $d/a$ for transverse (black solid line) and longitudinal (blue dashed line) polarizations.

Figure 5

Spectral heat flux decomposed in (a) and (c) transverse and (b) and (d) longitudinal polarizations for (a) and (b) $d=4a=100\mathrm{nm}$ and (c) and (d) $d=7a=175\mathrm{nm}$. In all the plots, the solid lines correspond to the Landauer approach, while the dashed lines are associated with the Boltzmann approach.

Figure 6

(a) Total flux as a function of distance calculated using Landauer's approach (black solid line) and Boltzmann's approach in the diffusive regime (blue dashed line) and in the ballistic regime (red dot-dashed line). The fluxes are renormalized with respect to the maximum flux given in Eq. (28). (b) Ratio between the total flux calculated using Landauer's approach and the Boltzmann diffusive one as a function of the distance.

Figure 7

Dispersion relation for a chain of Au nanoparticles with radius $a=5\mathrm{nm}$ and distance $d=4a=20\mathrm{nm}$ for transverse (black curve) and longitudinal (blue curve) polarizations. The lines correspond to the search for complex frequency $\omega$ (the plot shows the real part of $\omega$) using real $k$. The red vertical line is the light line in vacuum. Inset: Zoom into the region of strong coupling between the transversal chain modes and the radiation modes.

Figure 8

Propagation length for transverse (black) and longitudinal (blue) polarizations for a chain of Au nanoparticles with radius $a=5$ nm. The propagation length is particularly large for the frequency where the transversal chain modes couple strongly to the radiation modes.

Figure 9

Landauer result $P_e$ for a chain of Au nanoparticles with radius $a=5$ nm normalized to $P_{\max }$ from Eq. (28). Furthermore, we show the electric and magnetic contributions to $P_e$. The result from the kinetic approach (not shown here) has only an electric contribution, tens of orders of magnitude lower than the exact result.

Figure 10

Spectral heat flux for the electric contribution decomposed in transverse and longitudinal polarizations for $d=4a=20\mathrm{nm}$. In both plots, the solid black lines correspond to the Landauer approach, while the solid red lines are associated with the Boltzmann approach.