Abstract
Heat transfer in solids is modeled in the framework of kinetic theory of the phonon gas. The microscopic description of the phonon gas relies on the phonon Boltzmann equation and the Callaway model for phonon–phonon interaction. A simple model for phonon interaction with crystal boundaries, similar to the Maxwell boundary conditions in classical kinetic theory, is proposed. Macroscopic transport equation for an arbitrary set of moments is developed and closed by means of Grad’s moment method. Boundary conditions for the macroscopic equations are derived from the microscopic model and the Grad closure. As example, sets with 4, 9, 16, and 25 moments are considered and solved analytically for one-dimensional heat transfer and Poiseuille flow of phonons. The results show the influence of Knudsen number on phonon drag at solid boundaries. The appearance of Knudsen layers reduces the net heat conductivity of solids in rarefied phonon regimes.
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References
Romano V., Rusakov A.: 2d numerical simulations of an electron-phonon hydrodynamical model based on the maximum entropy principle. Comput. Methods Appl. Mech. Eng. 199, 2741–2751 (2010)
Kittel C.: Introduction to Solid State Physics, 7th edn. John Wiley & Sons, Hoboken (1996)
Snoke D.W.: Solid State Physics Essential Concepts. Addison-Wesley, San Francisco (2009)
Peraud J-P.M., Hadjiconstantinou N.G.: Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations. Phys. Rev. B 84, 205331 (2011)
Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)
Dreyer W., Struchtrup H.: Heat pulse experiments revisited. Contin. Mech. Thermodyn. 5, 3–50 (1993)
Alvarez F.X., Jou D., Sellitto A.: Phonon hydrodynamics and phonon-boundary scattering in nanosystems. J. Appl. Phys. 105, 014317 (2009)
Sellitto A., Alvarez F.X., Jou D.: Second law of thermodynamics and phonon-boundary conditions in nanowires. J. Appl. Phys. 107, 064302 (2010). doi:10.1063/1.3309477
Alvarez F.X., Cimmelli V.A., Jou D., Sellitto A.: Mesoscopic description of boundary effects in nanoscale heat transport. Nanoscale Syst. MMTA 1, 112–142 (2012)
Guyer R.A., Krumhansl J.A.: Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, 766–778 (1966)
Guyer R.A., Krumhansl J.A.: Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778–788 (1966)
Callaway J.: Model for lattice thermal conductivity at low temperatures. Phys. Rev. 113, 1046–1051 (1959)
Peierls R.: Zur kinetischen Theorie der Wärmeleitung in Kristallen. Annalen der Physik 3, 1055–1101 (1929)
Debye P.: Zur Theorie der spezifischen Wärmen. Annalen der Physik 39, 789–839 (1912)
Einstein A.: Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme. Annalen der Physik 22, 180–190 (1907)
Klemens P.G.: Anharmonic decay of optical phonons. Phys. Rev. 148, 845–848 (1966)
Waldmann L.: Transporterscheinungen in Gasen von mittlerem Druck. In: Flügge, S. (ed.) Handbuch der Physik XII: Thermodynamik der Gase, Springer, Berlin (1958)
Struchtrup H.: Macroscopic Transport Equations for Rarefied Gas Flows. Springer, Heidelberg (2005)
Müller I.: Thermodynamics. Pitman Publishing, Boston (1985)
Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)
Fryer, M.: The Macroscopic Transport Equations of Phonons in Solids. MASc thesis, University of Victoria (2012)
Torrilhon M., Struchtrup H.: Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227, 1982–2011 (2008)
Struchtrup H., Torrilhon M.: Higher-order effects in rarefied channel flows. Phys. Rev. E 78, 046301 (2008)
Hadjiconstantinou N.G.: Comment on Cercignani’s second-order slip coefficient. Phys. Fluids 15, 2352–2354 (2003)
Struchtrup H.: Linear kinetic heat transfer: moment equations, boundary conditions, and Knudsen layers. Physica A 387, 1750–1766 (2008)
Taheri P., Torrilhon M., Struchtrup H.: Couette and Poiseuille flows in microchannels: analytical solutions for regularized 13-moment equations. Phys. Fluids 21, 017102 (2009)
Johnson J.A., Maznev A.A., Cuffe J., Eilason J.K., Minnich A.J., Kehoe T., Sotomayor Torres C.M., Chen G., Nelson K.A.: Direct measurement of room temperature non-diffusive thermal transport over micron distance in a silicon membrane. Phys. Rev. Lett. 110, 025901 (2013)
Struchtrup H., Taheri P.: Macroscopic transport models for rarefied gas flows: a brief review. IMA J. Appl. Math. 76, 672–697 (2011)
Struchtrup H., Torrilhon M.: Regularized 13 moment equations for hard sphere molecules: linear bulk equations. Phys. Fluids 25, 052001 (2013)
Rana A.S., Torrilhon M., Struchtrup H.: A robust numerical method for the R13 equations of rarefied gas dynamics: application to lid driven cavity. J. Comput. Phys. 236, 169–186 (2013)
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Communicated by Andreas Öchsner.
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Fryer, M.J., Struchtrup, H. Moment model and boundary conditions for energy transport in the phonon gas. Continuum Mech. Thermodyn. 26, 593–618 (2014). https://doi.org/10.1007/s00161-013-0320-y
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DOI: https://doi.org/10.1007/s00161-013-0320-y