# Heat flux at the nanoscale : beyond the Boltzmann-Stefan law

Does the usual Stephan-Boltzmann theory for blackbody radiation applies to nanometer-size objects ? To answer this question the heat flux in vacuum between two surfaces at different temperature and separated by distances between a micrometer to100 nm have been measured and compared to theory by two CNRS labs (Charles Fabry of Institut d’Optique and Institut Néel). At the nanometer scale, the measurements show large discrepancies with the Stefan-Boltzmann theory which describes this thermal exchange at large distances on the basis of Planck’s law. As well known, in the far field regime the heat flux exchanged between two flat parallel surfaces does not depend upon the distance between the two surfaces. Instead, in the near field regime, the measured variation is strong. The flux increase dramatically as the distance between the two surfaces becomes smaller than about one micrometer.

In fact, this effect had been first discovered in the framework of the Apollo program, as efficient protections against radiation were needed. An increase of the heat flux between two metallic films was observed with very high values. In 1971, Polder and van Hove made a complete theoretical study of this phenomenon. Stimulated by recent advances in NEMS, for which these increased heat flux at short distances can be of importance, an experimental set up has been specifically designed. Experimental developments in the measurement of the Casimir force under vacuum [3] guided the setup design : the improvements in measuring techniques coming from Scanning Probe Microscopy and availability of new techniques to fabricate micro-objects have given access to the experimental study of the sub-micrometer regime of the radiative heat flux. Already in 2006, the thermal radiation has been detected in the near field regime [4] using these techniques. Second, the theory group at the Institut d’Optique [5] has been able to produce a quantitative calculation between polar dielectric surfaces like glass substrates used here and has shown that the expected flux is here larger by a factor of 10 compared to metallic surfaces. This is essentially due to the resonant excitation of the surface phonon polaritons that are present in glasses. |
Fig. 1 : Thermal conductance between the sphere with diameter 40 micrometers and the plate as a function of gap distance.The black dots represent experimental data and the red line the theoretical model. The temperature difference between the plate and the sphere is 21 K.
The underlying physics governing the large increased of the heat flux at the nanometer scale shown in Fig 1 is closely related to the Casimir effect. This mechanical effect, an attractive force between two mirrors, is due to the quantum fluctuations in the electrodynamic coupling between these mirrors. Instead of quantum fluctuations, the thermal coupling between the two surfaces through overlapping of evanescent waves is due to thermally induced electron surface excitations. Here, these thermally surface excitations are phonon-polariton waves which can be thermally excited at low temperature. This can be accounted by the dipolar coupling between two nanoparticles separated by nanometer distances. This dipolar coupling determines the attractive van der Waals interaction between these particles and their energy exchange. The dipolar coupling is not taken into account in Planck theory in the far field theory. It becomes the dominant contribution at short distances. The precision obtained in this measurement opens the possibility to enter experimental studies of roughness effects or of non local effects at distances much smaller than a micrometer. For a more efficient comparison between theory and experiment, an experimental vacuum force machine adapted to a parallel plane geometry is being prepared. |

**Further readings :**

[1] Radiative heat transfer at the nanoscale, E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier and J.J. Greffet, Nature Photonics 3, 514 - 517 (2009)

[2] Quantitative non-contact dynamic Casimir force measurements G. Jourdan, A. Lambrecht, F. Comin and J. Chevrier EPL 85 31001 (2009)

[3] Thermal Radiation Scanning Tunnelling Microscopy Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.A. Lemoine, K. Joulain, J.P. Mulet, Y. Chen and J.J. Greffet, 444, p 740 Nature (2006).

[4] Nanoscale radiative heat transfer between a small particle and a plane surface » J.P. Mulet, K. Joulain, R. Carminati and J.J. Greffet, Appl. Phys. Lett. 78, 2931-2933 (2001).

*Cet article est repris du site* C’NANO Rhône-Alpes