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Nonlocal heat conduction in silicon nanowires and carbon nanotubes

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Abstract

Experimental results showed that the effective thermal conductivity of silicon nanowire is smaller than the bulk thermal conductivity, while that of carbon nanotube (CNT) is usually much larger than its bulk counterpart. In order to resolve this paradox, a nonlocal heat conduction model for one-dimensional materials is proposed. This nonlocal model indicates that the different heat conduction boundary conditions of silicon nanowire and CNT lead to the different behaviour of their thermal conductivities in comparison with their bulk counterparts. Furthermore, the nonlocal effect of heat flux on the surfaces of the CNT makes the thermal conductivity of the single-wall CNT more than seven orders of magnitude higher than its bulk thermal conductivity. The thermal conductivities of the single-wall and multi-wall CNTs obtained by using the nonlocal model show an agreement with the experimental ones.

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References

  1. Lepri S, Livi R, Politi A (2003) Thermal conduction in classical low-dimensional lattices. Phys Rep 377:1–80

    MathSciNet  Google Scholar 

  2. Balandin AA (2009) Chill out: new materials and designs can keep chips cool. IEEE Spectr 46:34–39

  3. Narayan O, Ramaswamy S (2002) Anomalous heat conduction in one-dimensional momentum-conserving systems. Phys Rev Lett 89:200601

    Google Scholar 

  4. Basile G, Bernardin C, Olla S (2006) Momentum conserving model with anomalous thermal conductivity in low dimensional systems. Phys Rev Lett 96:204303

    Google Scholar 

  5. Yang L, Grassberger P, Hu B (2006) Dimensional crossover of heat conduction in low dimension. Phys Rev E 74:062101

    Google Scholar 

  6. Chang CW et al (2008) Breakdown of Fourier’s law in nanotube thermal conductors. Phys Rev Lett 101:075903

    Google Scholar 

  7. Henry A, Chen G (2008) High thermal conductivity of single polyethylene chains using molecular dynamic simulations. Phys Rev Lett 101:235502

    Google Scholar 

  8. Jou D, Criado-Sancho M, Casas-Vázquez J (2010) Heat fluctuations and phonon hydrodynamics in nanowires. J Appl Phys 107:084302

    Google Scholar 

  9. Sellitto A, Alvarez FX, Jou D (2010) Second law of thermodynamics and phonon-boundary conditions in nanowires. J Appl Phys 107:064302

    Google Scholar 

  10. Yang N, Zhang G, Li B (2010) Violation of Fourier’s law and anomalous heat diffusion in silicon nanowires. Nano Today 5:85–90

    Google Scholar 

  11. Ghosh S et al (2010) Dimensional crossover of thermal transport in few-layer graphene. Nat Mater 9:555–558

    Google Scholar 

  12. Liu J, Yang R (2012) Length-dependent thermal conductivity of single extended polymer chains. Phys Rev B 86:104307

    Google Scholar 

  13. Xu X et al (2014) Length-dependent thermal conductivity in suspended single layer graphene. Nat Commun 5:3689

    Google Scholar 

  14. Sellitto A, Carlomagno I, Jou D (2015) Two-dimensional phonon hydrodynamics in narrow strips. Proc Royal Soc A: Math Phys Eng Sci 471:20150376

    MathSciNet  MATH  Google Scholar 

  15. Carlomagno I, Sellitto A, Jou D (2015) Effective phonon mean-free path and slip heat flow in rarefied phonon hydrodynamics. Phys Lett A 379:2652–2656

    Google Scholar 

  16. Sonvane Y et al (2015) Length, width and roughness dependent thermal conductivity of graphene nanoribbons. Chem Phys Lett 634:16–19

    Google Scholar 

  17. Cepellotti A et al (2015) Phonon hydrodynamics in two-dimensional materials. Nat Commun 6:6400

    Google Scholar 

  18. Lee S, Broido D, Esfarjani K, Chen G (2015) Hydrodynamic phonon transport in suspended graphene. Nat Commun 6:6290

    Google Scholar 

  19. Wang M, Yang N, Guo ZY (2011) Non-Fourier heat conductions in nanomaterials. J Appl Phys 110:064310

    Google Scholar 

  20. Wang M, Guo ZY (2010) Understanding of temperature and size dependences of effective thermal conductivity of nanotubes. Phys Lett A 374:4312–4315

    MATH  Google Scholar 

  21. Torres P (2018) Emergence of hydrodynamic heat transport in semiconductors at the nanoscale. Phys Rev Mater 2:076001

    Google Scholar 

  22. Cattaneo C (1958) A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247:431–433

    MATH  Google Scholar 

  23. Vernotte P (1961) Some possible complications in the phenomena of thermal conduction. Compte Rendus 252:2190–2191

    Google Scholar 

  24. Tzou DY (1996) Macro- to microscale heat transfer: the lagging behavior. Taylor & Francis, Washington

    Google Scholar 

  25. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41

    MathSciNet  MATH  Google Scholar 

  26. Cheng L, Xu MT, Wang LQ (2008) From Boltzmann transport equation to single-phase-lagging heat conduction. Int J Heat Mass Transf 51:6018

    MATH  Google Scholar 

  27. Xu MT, Wang LQ (2005) Dual-phase-lagging heat conduction based on Boltzmann transport equation. Int J Heat Mass Transf 48:5616

    MATH  Google Scholar 

  28. Cheng L, Xu MT, Wang LQ (2008) Thermal vibration phenomenon of single phase lagging heat conduction and its thermodynamic basis. Chin Sci Bull 53:3597

    MathSciNet  Google Scholar 

  29. Xu MT, Wang LQ (2002) Thermal oscillation and resonance in dual-phase-lagging heat conduction. Int J Heat Mass Transf 45:1055

    MATH  Google Scholar 

  30. Fujii M et al (2005) Measuring the thermal conductivity of a single carbon nanotube. Phys Rev Lett 95:065502

    Google Scholar 

  31. Ju YS, Goodson KE (1999) Phonon scattering in silicon films with thickness of order 100nm. Appl Phys Lett 74:3005

    Google Scholar 

  32. Liu W, Asheghi M (2004) Phonon-boundary scattering in ultrathin single-crystal silicon layers. Appl Phys Lett 84:3819

    Google Scholar 

  33. Li D et al (2003) Thermal conductivity of individual silicon nanowires. Appl Phys Lett 83:2934

    Google Scholar 

  34. Asheghi M et al (2002) Thermal conduction in doped single-crystal silicon films. J Appl Phys 91:5079

    Google Scholar 

  35. Guyer RA, Krumhansl JA (1966) Solution of the linearized phonon Boltzmann equation. Phys Rev 148:766

    Google Scholar 

  36. Guyer RA, Krumhansl JA (1966) Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals. Phys Rev 148:778

    Google Scholar 

  37. Jou D, Casas-Vázquez J (1990) Nonequilibrium absolute temperature, and phonon hydrodynamics. Physica A 163:47

    Google Scholar 

  38. Guo Y, Wang M (2015) Phonon hydrodynamics and its applications in nanoscale heat transport. Phys Rep 595:1–44

    MathSciNet  Google Scholar 

  39. Sobolev SL (1993) Two-temperature discrete model for nonlocal heat conduction. J Phys III France 3:2261–2269

    Google Scholar 

  40. Sobolev SL (1994) Equations of transfer in nonlocal media. Int J Heat Mass Transf 37:2175–2182

    MATH  Google Scholar 

  41. Sobolev SL (2016) Nonlocal two-temperature model: application to heat transport in metals irradiated by ultrashort laser pulses. Int J Heat Mass Transf 94:138–144

    Google Scholar 

  42. Jou D, Cimmelli VA, Sellitto A (2012) Nonlocal heat transport with phonons and electrons: application to metallic nanowires. Int J Heat Mass Transf 55:2338–2344

    MATH  Google Scholar 

  43. Jou D, Casas-Vázquez J, Lebon G (2010) Extended irreversible thermodynamics, fourth ed. Springer-Verlag, Berlin

    MATH  Google Scholar 

  44. Alvarez FX, Jou D, Sellitto A (2011) Phonon boundary effects and thermal conductivity of rough concentric nanowires. J Heat Transf 133:022402

    Google Scholar 

  45. Lebon G (2014) Heat conduction at micro and nanoscales: a review through the prism of extended irreversible thermodynamics. J Non-Equilib Thermodyn 39:35–59

    Google Scholar 

  46. Cimmelli VA (2009) Different thermodynamic theories and different heat conduction laws. J Non-Equilib Thermodyn 34:299–333

    MATH  Google Scholar 

  47. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710

    Google Scholar 

  48. Romano G, Barretta R, Diaco M (2017) On nonlocal integral models for elastic nano-beams. Int J Mech Sci 131-132:490–499

    Google Scholar 

  49. Romano G et al (2017) Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int J Mech Sci 121:151–156

    Google Scholar 

  50. Cahill DG et al (2014) Nanoscale thermal transport. II. 2003-2013. Appl Phys Rev 1:011305

    Google Scholar 

  51. Klemens PG (2001) Theory of thermal conduction in thin ceramic films. Int J Thermophys 22(1):265–275

    Google Scholar 

  52. Gill-Comeau M, Lewis LJ (2015) Heat conductivity in graphene and related materials: a time-domain model analysis. Phys Rev B 92:195404

    Google Scholar 

  53. Lee V et al (2017) Divergent and ultrahigh thermal conductivity in millimeter-long nanotubes. Phys Rev Lett 118:135901

    Google Scholar 

  54. Cimmelli VA, Sellitto A, Jou D (2010) Propagation of temperature waves along core-shell nanowires. J Non-Equilib Thermodyn 35:267–278

    MATH  Google Scholar 

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Acknowledgements

The financial support of our research by National Natural Science Foundation of China (Project No. 50876054) is greatly appreciated.

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  1. Department of Engineering Mechanics, Shandong University, Jinan, 250061, China

    Mingtian Xu

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  1. Mingtian Xu
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Correspondence to Mingtian Xu.

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Xu, M. Nonlocal heat conduction in silicon nanowires and carbon nanotubes. Heat Mass Transfer 57, 843–852 (2021). https://doi.org/10.1007/s00231-020-02994-8

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  • DOI: https://doi.org/10.1007/s00231-020-02994-8

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