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Physical Mesomechanics

, Volume 20, Issue 3, pp 305–310 | Cite as

Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

  • A. A. SokolovEmail author
  • A. M. Krivtsov
  • W. H. Müller
Article

Abstract

In this paper exact analytical solutions for the equation that describes anomalous heat propagation in a harmonic 1D lattices are obtained. Rectangular, triangular and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to \(1/\sqrt t \). In the center of the perturbation zone the decay is proportional to 1/t. Thus, the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experimentally.

Keywords

heat conduction harmonic crystals one dimensional crystals localized excitations anomalous heat conduction 

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References

  1. 1.
    Zhang, X., Luo, D., Cui, G., Wang, Y., and Huang, B., Construction of Logic Gate Based on Multi-Channel Carbon Nanotube Field-Effect Transistors, Proc. of III Int. Conf. Intelligent Human-Machine Systems and Cybernetics, Aug. 2011, 2011, vol. 2, pp. 94–97.ADSGoogle Scholar
  2. 2.
    Chen, Ch., Jin, T., Wei, L., Li, Y., Liu, X., Wang, Y., Zhang, L., Liao, Ch., Hu, N., Song, Ch., and Zhang, Y., High-Work-Function Metal/Carbon Nanotube/Low-Work-Function Metal Hybrid Junction Photovoltaic Device, NPG Asia Mater., 2015, vol. 7, p. e220.CrossRefGoogle Scholar
  3. 3.
    Goldstein, R.V. and Morozov, N.F., Mechanics of Deformation and Fracture of Nanomaterials and Nanotechnology, Phys. Mesomech., 2007, vol. 10, no. 5-6, pp. 235–246.CrossRefGoogle Scholar
  4. 4.
    Li, B., Wang, L., and Casati, G., Thermal Diode: Rectification of Heat Flux, Phys. Rev. Lett., 2004, vol. 93, p. 184–301.Google Scholar
  5. 5.
    Brown, E., Hao, L., Gallop, J.C., and Macfarlane, J.C., Ballistic Thermal and Electrical Conductance Measurements on Individual Multiwall Carbon Nanotubes, Appl. Phys. Lett., 2005, vol. 87, no. 2, p. 023107.ADSCrossRefGoogle Scholar
  6. 6.
    Wang, Zh., Carter, J.A., Lagutchev, A., KanKoh, Y., Seong, N.-H., Cahill, D.G., and Dlott, D.D., Ultrafast Flash Thermal Conductance of Molecular Chains, Science, 2007, vol. 317, no. 5839, pp. 787–790.ADSCrossRefGoogle Scholar
  7. 7.
    Cannon, J.R., The One-Dimensional Heat Equation, Cambridge: Cambridge University Press, 1984.CrossRefzbMATHGoogle Scholar
  8. 8.
    Lepri, S., Livi, R., and Politi, A., Thermal Conduction in Classical Low-Dimensional Lattices, Phys. Rep., 2003, pp. 1–80.Google Scholar
  9. 9.
    Hsiao, T.-K., Chang, H.-K., Liou, Sz-Ch., Chu, M.-W., Lee, Si-Ch., and Chang, Ch.-W., Observation of Room-Temperature Ballistic Thermal Conduction Persisting over 8.3 jam in SiGe Nanowires, Nat. Nanotech., 2013, vol. 8, no. 7, pp. 534–538.ADSCrossRefGoogle Scholar
  10. 10.
    Zhang, H., Hua, Ch., Ding, D., and Minnich, A.J., Length Dependent Thermal Conductivity Measurements Yield Phonon Mean Free Path Spectra in Nanostructures, Sci. Rep., 2015, vol. 5, p. 9121.ADSCrossRefGoogle Scholar
  11. 11.
    Chang, C.W., Okawa, D., Garcia, H., Majumdar, A., and Zettl, A., Breakdown of Fourier’s Law in Nanotube Thermal Conductors, Phys. Rev. Lett., 2008, vol. 101, p. 075903.ADSCrossRefGoogle Scholar
  12. 12.
    Rieder, Z., Lebowitz, J.L., and Lieb, E., Properties of a Harmonic Crystal in a Stationary Nonequilibrium State, J. Math. Phys., 1967, vol. 8, no. 5. pp. 1073–1078.ADSCrossRefGoogle Scholar
  13. 13.
    Cattaneo, C., Sur une Forme de L’équation de la Chaleur Eliminant le Paradoxe d’ une Propagation Instantanee, Comptes Rendus, 1958, vol. 247, pp. 431–433.zbMATHGoogle Scholar
  14. 14.
    Vernotte, P., Les Paradoxes de la Theorie Continue de L’équation de la Chaleur, Comptes Rendus, 1958, vol. 246, pp. 3154–3155.zbMATHGoogle Scholar
  15. 15.
    Gendelman, O.V. and Savin, A.V., Nonstationary Heat Conduction in One-Dimensional Chains with Conserved Momentum, Phys. Rev. E, 2010, vol. 81, p. 020103.ADSCrossRefGoogle Scholar
  16. 16.
    Krivtsov, A.M., Energy Oscillations in a One-Dimensional Crystal, Dokl. Phys., 2014, vol. 59, no. 9, pp. 427–430.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Krivtsov, A.M., Heat Transfer in Infinite Harmonic One Dimensional Crystals, Dokl. Phys., 2015, vol. 60, no. 9, pp. 407–411.ADSCrossRefGoogle Scholar
  18. 18.
    Krivtsov, A.M., On Unsteady Heat Conduction in a Harmonic Crystal, ArXivPreprint, 2015, p. 1509.02506.Google Scholar
  19. 19.
    Babenkov, M.B., Krivtsov, A.M., and Tsvetkov, D.V., Energy Oscillations in a One-Dimensional Harmonic Crystal on an Elastic Substrate, Phys. Mesomech., 2016, vol. 19, no. 3, pp. 282–290.CrossRefGoogle Scholar
  20. 20.
    Kuzkin, V.A. and Krivtsov, A.M., An Analytical Description of Transient Thermal Processes in Harmonic Crystals, Phys. Solid State, 2017, vol. 59, no. 5, pp. 1051–1062.ADSCrossRefGoogle Scholar
  21. 21.
    Poletkin, K.V., Gurzadyan, G.G., Shang, J., and Kulish, V., Ultrafast Heat Transfer on Nanoscale in Thin Gold Films, Appl. Phys. B, 2012, vol. 107, no. 1, pp. 13–143.CrossRefGoogle Scholar
  22. 22.
    Polyanin, A.D. and Nazaikinskii, V.E., Handbook of Linear Partial Differential Equationsfor Engineers and Scientists, Boca Raton: Chapman and Hall/CRC, 2016.CrossRefzbMATHGoogle Scholar
  23. 23.
    Müller, I. and Müller, W.H., Fundamentals of Thermodynamics and Applications with Historical Annotations and Many Citations from Avogadro to Zermelo, Berlin: Springer, 2011.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • A. A. Sokolov
    • 1
    Email author
  • A. M. Krivtsov
    • 1
    • 2
  • W. H. Müller
    • 3
  1. 1.Peter the Great Saint-Petersburg Polytechnic UniversitySt.-PetersburgRussia
  2. 2.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSt.-PetersburgRussia
  3. 3.Institute of Mechanics, Chair of Continuum Mechanics and Constitutive TheoryTechnische Universität BerlinBerlinGermany

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