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Thermal resistance at a solid/superfluid helium interface

Abstract

Kapitza1 in 1941 discovered that heat flowing across a solid in contact with superfluid helium (<2 K) encounters a strong thermal resistance at the interface. Khalatnikov2 demonstrated theoretically that this constitutes a general phenomenon related to all interfaces at all temperatures, given the dependence of heat transmission on the acoustic impedance (sound velocity × density) of each medium. For the solid/superfluid interface, the measured transmission of heat is almost one hundred times stronger than the Khalatnikov prediction. This discrepancy could be intuitively attributed to diffuse scattering of phonons3 at the interface but, despite several attempts4,5,6,7, a detailed quantitative comparison between theoretical and experimental findings to explain the occurrence of scattering and its contribution to heat transmission had been lacking. Here we show that when the thermal wavelength λ of phonons of the less dense medium (liquid 4He) becomes comparable to the r.m.s. surface roughness σ, the heat flux crossing the interface is amplified; in particular when σ ≈ 0.33λ, a spatial resonant mechanism occurs, as proposed by Adamenko and Fuks8. We used a silicon single crystal whose surface roughness was controlled and characterized. The thermal boundary resistance measurements were performed from 0.4 to 2 K at different superfluid pressures ranging from saturated vapour pressure (SVP) to above 4He solidification, to eliminate all hypothetical artefact mechanisms. Our results demonstrate the physical conditions necessary for resonant phonon scattering to occur at all interfaces, and therefore constitute a benchmark in the design of nanoscale devices9,10 for heat monitoring.

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Figure 1: Schematic representation of the experimental cell.
Figure 2: Kapitza resistance RK measurements.
Figure 3: Surface roughness σ dependence on the dominant thermal wavelength λ (in nm).
Figure 4: Interaction probability Pφ(h) for a phonon of wavelength λ to undergo resonant scattering from a roughness of height h.
Figure 5: Typical power spectral density (PSD) of our sample surface as a function of scale length d.

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References

  1. Kapitza, P. L. The study of heat transfer in helium II. J. Phys. 4, 181–210 (1941).

    Google Scholar 

  2. Khalatnikov, I. M. An Introduction to the Theory of Superfludity (Addisson-Wesley, 1989).

    Google Scholar 

  3. Swartz, E. T. & Pohl, R. O. Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668 (1989).

    Article  Google Scholar 

  4. Wyatt, A. F. G. & Crisp, G. N. Frequency of phonons emitted into liquid He by a solid. J. Phys. 39, C6–C244 (1978).

    Google Scholar 

  5. Little, W. A. Unimportance of surface roughness upon the Kapitza resistance. Phys. Rev. Lett. 123, 1909–1911 (1961).

    Google Scholar 

  6. Sheard, F., Bowley, R. M. & Toombs, G. A. Microscopic theory of the Kapitza resistance at a solid–liquid 4He interface. Phys. Rev. A 8, 3135–3145 (1973).

    Article  CAS  Google Scholar 

  7. Challis, L., Dransfeld, K. & Wilks, J. Heat transfer between solids and liquid helium II. Proc. R. Soc. Lond. 260, 31–46 (1961).

    Article  CAS  Google Scholar 

  8. Adamenko, I. N. & Fuks, I. M. Roughness and thermal resistance of the boundary between a solid and liquid helium. Sov. Phys. JETP 32, 1123–1129 (1971).

    Google Scholar 

  9. Wen, Y.-C. et al. Specular scattering probability of acoustic phonons in atomically flat interfaces. Phys. Rev. Lett. 103, 264301 (2009).

    Article  Google Scholar 

  10. Hochbaum, A. I. et al. Enhanced thermoelectric performance of rough silicon nanowires. Nature 451, 163–167 (2008).

    Article  CAS  Google Scholar 

  11. Pain, H. J. The Physics of Vibrations and Waves (John Wiley, 1983).

    Google Scholar 

  12. Stoner, R. J. & Maris, H. J. Kapitza conductance and heat flow between solids at temperatures from 50 to 300 K. Phy. Rev. B 48, 16373–16387 (1993).

    Article  CAS  Google Scholar 

  13. Nakayama, T. in Prog. Low Temp. Phys. Vol. XII, 115 (Elsevier, 1989).

    Google Scholar 

  14. Adamenko, I., Nemchenko, K. & Tanatarov, I. Transmission and reflection of phonons and rotons at the superfluid helium-solid interface. Phys. Rev. B 77, 174510 (2008).

    Article  Google Scholar 

  15. Adamenko, I. N. & Nemchenko, E. K. Heat Flow from solid to liquid He II due to inelastic processes of phonons interaction. J. Low Temp. Phys. 171, 266–272 (2012).

    Article  Google Scholar 

  16. Brooks, J. S. & Donnelly, R. J. The calculated thermodynamic properties of superfluid helium-4. J. Phys. Chem. Ref. Data 6, 51–104 (1977).

    Article  CAS  Google Scholar 

  17. Maldovan, M. Sound and heat revolutions in phononics. Nature 503, 209–217 (2013).

    Article  CAS  Google Scholar 

  18. Amrit, J. Impact of surface roughness temperature dependency on the thermal contact resistance between Si(111) and liquid 4He. Phys. Rev. B 81, 054303 (2010).

    Article  Google Scholar 

  19. Horcas, I. et al. WSxM: a software for scanning probe microscopy and a tool for nanotechnology. Rev. Sci. Instrum. 78, 1–8 (2007).

    Article  Google Scholar 

  20. Grilly, E. R. Pressure-volume-temperature relations in liquid and solid 4He. J. Low Temp. Phys. 11, 33–52 (1973).

    Article  CAS  Google Scholar 

  21. Balibar, S., Alles, H. & Parshin, Y. A. The surface of helium crystals. Rev. Mod. Phys. 77, 317–370 (2005).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This work received partial financial aid from the LabEx LaSIPS of the Paris-Saclay University. We express our deep gratitude to the Institute of Nuclear Physics in Orsay for technical support. A.R. benefited from a grant from the Ministry of Education via ED 543 MIPEGE. We acknowledge discussions with I. N. Adamenko.

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A.R. and J.A. ran the experiments. A.R., S.V. and J.A. analysed the data, performed the calculations and wrote the manuscript.

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Correspondence to Jay Amrit.

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The authors declare no competing financial interests.

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Ramiere, A., Volz, S. & Amrit, J. Thermal resistance at a solid/superfluid helium interface. Nature Mater 15, 512–516 (2016). https://doi.org/10.1038/nmat4574

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