Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

Allen, Philip B.

doi:10.1103/physrevb.97.134307

Cited by 13 publications

(10 citation statements)

References 57 publications

(64 reference statements)

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“…3 changes geometry. We show in the inset of cous heat equations predict a steeper-than-Fourier's law or non-linear temperature gradient that is reminiscent of that obtained in molecular dynamics simulations [88][89][90][91][92][93][94][95][96][97] and in explicit solutions of the LBTE [34].…”

Section: Case Studysupporting

confidence: 59%

Generalization of Fourier’s Law into Viscous Heat Equations

2020

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Heat conduction in dielectric crystals originates from the propagation of atomic vibrational waves, whose microscopic dynamics is well described by linearized or generalized phonon Boltzmann transport. Recently, it was shown that the thermal conductivity can be resolved exactly and in a closed form as a sum over relaxons, i.e. the collective phonon excitations that are eigenvectors of Boltzmann equation's scattering matrix [Cepellotti and Marzari, Phys. Rev. X 6, 041013 (2016)]. Relaxons have a well-defined parity and only odd relaxons contribute to the thermal conductivity. Here, we show that the complementary set of even relaxons determines another quantity -the thermal viscosity -that enters into the description of heat transport in the hydrodynamic regime, where dissipation of crystal momentum by Umklapp scattering phases out. We also show how the thermal viscosity and conductivity parametrize two novel viscous heat equations -two coupled equations for the local temperature and drift velocity fields -which represent the thermal counterpart of the Navier-Stokes equations of hydrodynamics in the linear, laminar regime. These viscous heat equations are derived from a coarse-graining of the linearized Boltzmann transport equation for phonons, and encompass both limits of Fourier's law or second sound for strong or weak Umklapp dissipation, respectively. Last, we introduce the Fourier deviation number, a dimensionless parameter that quantifies the steady-state deviations from Fourier's law due to hydrodynamic effects. We showcase these findings in a test case of a complex-shaped device made of silicon or diamond. This formulation generalizes rigorously Fourier's heat equation, and extends the reach of microscopic computational techniques to characterize the fundamental parameters governing heat conduction. * michele.simoncelli@epfl.ch ics. Sussmann and Thellung [12], starting from the linearized BTE (LBTE) in absence of momentumdissipating (Umklapp) phonon-phonon scattering events, derived mesoscopic equations in terms of temperature and phonon drift velocity, the thermal counterpart of pressure and fluid velocity in liquids. Further advances came from Gurzhi [13,14] and Guyer & Krumhansl [15,16] who, including the effect of weak crystal momentum dissipation, obtained equations for damped second sound and Poiseuille heat flow. Among early works, we also mention the discussions on phonon hydrodynamics using approaches different from the LBTE of Refs. [17,18]. While correctly capturing the qualitative features of phonon hydrodynamics, all the theoretical investigations mentioned above are heuristic, e.g. they assume simplified phonon dispersion relations (either power-law [13,14] or linear isotropic [12,15,16]), or neglect momentum dissipation. A more rigorous and general formulation -albeit valid only in the hydrodynamic regime of weak Umklapp scattering -was introduced by Hardy, who extended the discussion of second sound [19] and, together with Albers, of Poiseuille flow in terms of mesoscopic transport equations...

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Section: Case Studysupporting

confidence: 59%

Generalization of Fourier’s Law into Viscous Heat Equations

2020

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“…(2) is equivalent to a nonlocal kernel of the thermal conductivity described in Ref. [15]. The FMM given by Eq.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Indications of Phonon Hydrodynamics in Telescopic Silicon Nanowires

et al. 2019

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The validity of Fourier's law in telescopic nanowires is tested by means of molecular dynamics simulations. We observe that the radial dependence of the heat-flux profile, the temperature jump, and the appearance of vorticity obtained in molecular dynamics telescopic wires near the contact point acquire a hydrodynamic character, and we show that they are incompatible with Fourier's law. We propose the Guyer-Krumhansl equation as a generalization capable of capturing these hydrodynamic effects. Latticedynamics results show that the thermal average of the confined modes shows no radial dependence of the vibrating energy inside the wire. This means that hydrodynamic effects could not be related to the confinement effects in these small systems.

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“…dξ , is nonlocal, meaning the contribution at a given point is determined by convolving the heat source function with an exponential decay function with a decay length of μx . While the nonlocality of thermal conductivity was identified by earlier works on phonon transport [6,[11][12][13]20,39,43,54], the contribution from the inhomogeneous term has been neglected. Recently, Allen and Perebeinos [43] considered the effects of external heating and derived a thermal susceptibility based on the PBE that links external heat generation to temperature response and a thermal conductivity that links temperature response to heat flux.…”

Section: A Governing Equationmentioning

confidence: 99%

Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

et al. 2019

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Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerable interest for basic science and thermal management technologies. However, debate exists over the appropriate constitutive law that defines thermal conductivity in the nondiffusive regime. Here, we derive a generalized Fourier's law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and for general geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation. This generalized Fourier's law predicts that thermal conductivity not only becomes nonlocal at length scales smaller than phonon mean free paths but also requires the inclusion of an inhomogeneous nonlocal source term that has been previously neglected. We provide evidence for the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without the generalized Fourier's law can lead to inaccurate measurement of thermal transport properties.

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Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

Cited by 13 publications

References 57 publications

Generalization of Fourier’s Law into Viscous Heat Equations

Generalization of Fourier’s Law into Viscous Heat Equations

Indications of Phonon Hydrodynamics in Telescopic Silicon Nanowires

Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

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