Advertisement

The Ballistic Heat Equation for a One-Dimensional Harmonic Crystal

  • Anton KrivtsovEmail author
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 103)

Abstract

The analytical model of unsteady ballistic heat transfer in a one-dimensional harmonic crystal is analyzed. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-reversible equation for the kinetic temperature is derived and solved. This equation can be referred as the ballistic heat conduction equation, it is somewhat similar to the hyperbolic heat conduction equation, but it has important differences. The resulting constitutive law for the heat flux in the considered system is obtained. This law significantly differs from Fourier’s law and it predicts a finite velocity of the heat front and independence of the heat flux on the crystal length. The analytical results are confirmed by computer simulations. Further developments of the presented approach are referred.

Notes

Acknowledgements

The author is grateful to O. V. Gendelman, W. G. Hoover, D. A. Indeitsev, M. L. Kachanov, V. A. Kuzkin, S. A. Lurie, N. F. Morozov, and V. F. Zhuravlev for helpful and stimulating discussions; to M. B. Babenkov and D. V. Tsvetkov for additional analysis and computations. This work is supported by the Russian Science Foundation (Grant No. 18-11-00201).

References

  1. 1.
    Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer. In: Lepri S. (ed.) Lecture Notes in Physics, vol. 921, 418 p. Springer, Switzerland (2016)Google Scholar
  2. 2.
    Hoover, W.G., Hoover, C.G.: Simulation and control of chaotic nonequilibrium systems. In: Advanced Series in Nonlinear Dynamics, vol. 27, 324 p. World Scientific (2015)Google Scholar
  3. 3.
    Charlotte, M., Truskinovsky, L.: Lattice dynamics from a continuum viewpoint. J. Mech. Phys. Solids 60, 1508–1544 (2012)CrossRefGoogle Scholar
  4. 4.
    Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Fokas, A., et al. (eds.) Mathematical Physics 2000, pp. 128–150. Imperial College Press, London (2000)CrossRefGoogle Scholar
  5. 5.
    Lepri, S., Livi, R., Politi, A.: Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377, 1–80 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dhar, A.: Heat transport in low-dimensional systems. Adv. Phys. 57, 457–537 (2008)CrossRefGoogle Scholar
  7. 7.
    Aoki, K., Kusnezov, D.: Bulk properties of anharmonic chains in strong thermal gradients: non-equilibrium \(\phi ^4\) theory. Phys. Lett. A 265, 250–256 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gendelman, O.V., Savin, A.V.: Normal heat conductivity of the one-dimensional lattice with periodic potential. Phys. Rev. Lett. 84, 2381–2384 (2000)CrossRefGoogle Scholar
  9. 9.
    Giardina, C., Livi, R., Politi, A., Vassalli, M.: Finite thermal conductivity in 1D lattices. Phys. Rev. Lett. 84, 2144–2147 (2000)CrossRefGoogle Scholar
  10. 10.
    Gendelman, O.V., Savin, A.V.: Normal heat conductivity in chains capable of dissociation. Europhys. Lett. 106, 34004 (2014)CrossRefGoogle Scholar
  11. 11.
    Bonetto, F., Lebowitz, J.L., Lukkarinen, J.: Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs. J. Stat. Phys. 116, 783–813 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Le-Zakharov, A.A., Krivtsov, A.M.: Molecular dynamics investigation of heat conduction in crystals with defects. Dokl. Phys. 53, 261–264 (2008)CrossRefGoogle Scholar
  13. 13.
    Chang, C.W., Okawa, D., Garcia, H., Majumdar, A., Zettl, A.: Breakdown of fourier’s law in nanotube thermal conductors. Phys. Rev. Lett. 101, 075903 (2008)CrossRefGoogle Scholar
  14. 14.
    Xu, X., Pereira, L.F., Wang, Y., Wu, J., Zhang, K., Zhao, X., Bae, S., Bui, C.T., Xie, R., Thong, J.T., Hong, B.H., Loh, K.P., Donadio, D., Li, B., Ozyilmaz, B.: Length-dependent thermal conductivity in suspended single-layer graphene. Nat. Commun. 5, 3689 (2014)CrossRefGoogle Scholar
  15. 15.
    Hsiao, T.K., Huang, B.W., Chang, H.K., Liou, S.C., Chu, M.W., Lee, S.C., Chang, C.W.: Micron-scale ballistic thermal conduction and suppressed thermal conductivity in heterogeneously interfaced nanowires. Phys. Rev. B 91, 035406 (2015)CrossRefGoogle Scholar
  16. 16.
    Kannan, V., Dhar, A., Lebowitz, J.L.: Nonequilibrium stationary state of a harmonic crystal with alternating masses. Phys. Rev. E 85, 041118 (2012)CrossRefGoogle Scholar
  17. 17.
    Dhar, A., Dandekar, R.: Heat transport and current fluctuations in harmonic crystals. Physica A 418, 49–64 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rieder, Z., Lebowitz, J.L., Lieb, E.: Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys. 8, 1073–1078 (1967)CrossRefGoogle Scholar
  19. 19.
    Rubin, R.J.: Momentum autocorrelation functions and energy transport in harmonic crystals containing isotopic defects. Phys. Rev. 131, 964–989 (1963)CrossRefGoogle Scholar
  20. 20.
    Gendelman, O.V., Shvartsman, R., Madar, B., Savin, A.V.: Nonstationary heat conduction in one-dimensional models with substrate potential. Phys. Rev. E 85, 011105 (2012)CrossRefGoogle Scholar
  21. 21.
    Gusev, A.A., Lurie, S.A.: Wave-relaxation duality of heat propagation in Fermi-Pasta-Ulam chains. Mod. Phys. Lett. B 26, 1250145 (2012)CrossRefGoogle Scholar
  22. 22.
    Guzev, M.A.: The Fourier law for a one-dimensional crystal. Far East. Math. J. (1), 34–39 (2018)Google Scholar
  23. 23.
    Krivtsov, A.M.: Heat transfer in infinite harmonic one-dimensional crystals. Dokl. Phys. 60, 407–411 (2015)CrossRefGoogle Scholar
  24. 24.
    Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev 39, 355–376 (1986)CrossRefGoogle Scholar
  25. 25.
    Tzou, D.Y.: Macro- to Microscale Heat Transfer: The Lagging Behavior, 566 p. Wiley (2015)Google Scholar
  26. 26.
    Poletkin, K.V., Gurzadyan, G.G., Shang, J., Kulish, V.: Ultrafast heat transfer on nanoscale in thin gold films. Appl. Phys. B 107, 137–143 (2012)CrossRefGoogle Scholar
  27. 27.
    Indeitsev, D.A., Osipova, E.V.: A two-temperature model of optical excitation of acoustic waves in conductors. Dokl. Phys. 62(3), 136–140 (2017)CrossRefGoogle Scholar
  28. 28.
    Lepri, S., Mejia-Monasterio, C., Politi, A.: A stochastic model of anomalous heat transport: analytical solution of the steady state. J. Phys. A Math. Theor. 42, 025001, 15 p. (2009)Google Scholar
  29. 29.
    Lepri, S., Mejia-Monasterio, C., Politi, A.: Nonequilibrium dynamics of a stochastic model of anomalous heat transport. J. Phys. A: Math. Theor. 43, 065002, 22 p. (2010)Google Scholar
  30. 30.
    Delfini, L., Lepri, S., Livi, R., Mejia-Monasterio, C., Politi, A.: Nonequilibrium dynamics of a stochastic model of anomalous heat transport: numerical analysis. J. Phys. A Math. Theor. 43, 145001, 16 p. (2009)Google Scholar
  31. 31.
    Krivtsov, A.M.: Dynamics of energy characteristics in one-dimensional crystal. In: Procedings of XXXIV Summer School “Advanced Problems in Mechanics”, St.-Petersburg, Russia, 2007, pp. 261–273. ISBN 5-98883-009-9Google Scholar
  32. 32.
    Krivtsov, A.M.: Energy oscillations in a one-dimensional crystal. Dokl. Phys. 59, 427–430 (2014)CrossRefGoogle Scholar
  33. 33.
    Krivtsov, A.M.: On unsteady heat conduction in a harmonic crystal (2015). ArXiv:1509.02506
  34. 34.
    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices, 432 p. Clarendon Press, Oxford (1954)Google Scholar
  35. 35.
    Gavrilov, S.N., Krivtsov, A.M., Tsvetkov, D.V.: Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply. In: Continuum Mechanics and Thermodynamics (2018)Google Scholar
  36. 36.
    Manvi, R., Duvall, G.E., Lowell, S.C.: Finite amplitude longitudinal waves in lattices. Int. J. Mech. Sci. 11, 1 (1969)CrossRefGoogle Scholar
  37. 37.
    Holian, B.L., Straub, G.K.: Molecular dynamics of shock waves in one-dimensional chains. Phys. Rev. B 18, 1593 (1978)CrossRefGoogle Scholar
  38. 38.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1046 p. U. S. Government Printing Office (1972)Google Scholar
  39. 39.
    Polyanin, A.D., Nazaikinskii, V.E.: Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd edn, 1632 p. CRC Press, Boca Raton-London (2016)Google Scholar
  40. 40.
    Babenkov, M.B., Krivtsov, A.M., Tsvetkov, D.V.: Unsteady heat conduction processes in a harmonic crystal with a substrate potential. In: Proceedings of XXIV ICTAM, 21–26 August 2016, Montreal, Canada, 2440–2441 (2016)Google Scholar
  41. 41.
    Shoby, K., Yoshida, T., Mori, H.: Dynamic properties of one-dimensional harmonic liquids. II, Prog. Theoret. Phys. 66, 1160–1168 (1981)Google Scholar
  42. 42.
    Kuzkin, V.A., Krivtsov, A.M.: Fast and slow thermal processes in harmonic scalar lattices. J. Phys. Condens. Matter 29, 505401, 14 p. (2017)Google Scholar
  43. 43.
    Kuzkin, V.A.: Fast and slow thermal processes in harmonic crystals with polyatomic lattice. Arxiv preprint(2018). ArXiv:1808.00504
  44. 44.
    Hoover, W.G.: Molecular dynamics. In: Lecture Notes in Physics, vol. 258, 324 p. Springer-Verlag (1986)Google Scholar
  45. 45.
    Holian, B.L., Hoover, W.G., Posch, H.A.: Resolution of Loschmidt’s paradox: the origin of irreversible behavior in reversible atomistic dynamics. Phys. Rev. Lett. 59, 10–13 (1987)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Gendelman, O.V., Savin, A.V.: Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys. Rev. E 81, 020103 (2010)CrossRefGoogle Scholar
  47. 47.
    Dhar, A.: Heat conduction in the disordered harmonic chain revisited. Phys. Rev. Lett. 86, 5882–5885 (2001)CrossRefGoogle Scholar
  48. 48.
    Hoover, W.G., Hoover, C.G.: Hamiltonian thermostats fail to promote heat flow. Commun. Nonlinear Sci. Numer. Simulat. 18, 3365–3372 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Kachman, T.: Master-degree thesis under supervision of O. V. Gendelman, 187 p., Technion, Izrael (2011)Google Scholar
  50. 50.
    Tzou, D.Y.: The generalized lagging responce in small scale and high rate heating. Int. J. Mass Transf. 38, 3231–3240 (1995)CrossRefGoogle Scholar
  51. 51.
    Rubin, R.J., Greer, W.L.: Abnormal lattice thermal conductivity of a one-dimensional, harmonic, isotopically disordered crystal. J. Math. Phys. 12, 1686–1701 (1971)CrossRefGoogle Scholar
  52. 52.
    Babenkov, M.B., Ivanova, E.A.: Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type. Contin. Mech. Thermodyn. 26, 483–502 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Sokolov, A.A., Krivtsov, A.M., Müller, W.H.: Localized heat perturbation in harmonic 1D crystals: solutions for an equation of anomalous heat conduction. Phys. Mesomech. 20(3), 305–310 (2017)CrossRefGoogle Scholar
  54. 54.
    Krivtsov, A.M., Kuzkin, V.A.: Discrete and continuum thermomechanics. In: Encyclopedia of Continuum Mechanics, 16 p (2018)Google Scholar
  55. 55.
    Goldstein, R.V., Morozov, N.F.: Mechanics of deformation and fracture of nanomaterials and nanotechnology. Phys. Mesomech. 10, 235–246 (2007)CrossRefGoogle Scholar
  56. 56.
    Shtukin, L.V., Berinskii, I.E., Indeitsev, D.A., Morozov, N.F., Skubov, D.Y.: Electromechanical models of nanoresonators. Phys. Mesomech. 19(3), 248–254 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Theoretical MechanicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Laboratory of Discrete Models in MechanicsInstitute for Problems in Mechanical Engineering RASSt. PetersburgRussia

Personalised recommendations