Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures

Guo, Yangyu; Wang, Moran

doi:10.1103/physrevb.97.035421

Cited by 86 publications

(93 citation statements)

References 82 publications

(193 reference statements)

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“…First, they are valid for general phonondispersion relations; previous mesoscopic approaches for hydrodynamic thermal transport replaced these with power-law or linear-isotropic relations [14][15][16], which are reasonable approximations only at cryogenic temperatures. In addition, we take into account the full collision matrix, refining other models derived from the LBTE in the SMA [70] or in the Callaway approximation [72]. It is worth noting that Hardy & Albers [20] derived a set of mesoscopic equations from the LBTE that may be regarded as the generalization of the Guyer-Krumhansl equation for a general phonon dispersion relation (i.e.…”

Section: Viscous Heat Equationsmentioning

confidence: 99%

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: Viscous Heat Equationsmentioning

confidence: 99%

Generalization of Fourier’s Law into Viscous Heat Equations

2020

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“…In summary, larger values of C lead to less resistance to heat flow and thus to greater values of the thermal Figure 4: The influence of non-uniform slip coefficients C on the normalised ETC k eff /k of a square nanowire (φ = 1). Lines correspond to the ETC computed using the slip coefficient in (22). The cases β = 0, β = 2, and β = −0.9 correspond slip coefficients that are constant, increasing towards a corner, and decreasing towards a corner.…”

Section: Resultsmentioning

confidence: 99%

“…3.4, the large-Kn behaviour of the ETC for a constant slip coefficient in the case of a square nanowire is given by k eff /k ∼ (C equiv /2)Kn −1 . By inserting the slip coefficient (22) into (20), performing the integration, and equating the result to (C equiv /2)Kn −1 , we find an equivalent slip constant given by…”

Section: Resultsmentioning

confidence: 99%

Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics

Calvo-Schwarzwälder

Hennessy

Torres

et al. 2018

International Journal of Heat and Mass Transfer

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A mathematical model is presented for thermal transport in nanowires with rectangular cross sections. Expressions for the effective thermal conductivity of the nanowire across a range of temperatures and cross-sectional aspect ratios are obtained by solving the Guyer-Krumhansl hydrodynamic equation for the thermal flux with a slip boundary condition. Our results show that square nanowires transport thermal energy more efficiently than rectangular nanowires due to optimal separation between the boundaries. However, circular nanowires are found to be even more efficient than square nanowires due to the lack of corners in the cross section, which locally reduce the thermal flux and inhibit the conduction of heat. By using a temperature-dependent slip coefficient, we show that the model is able to accurately capture experimental data of the effective thermal conductivity obtained from Si nanowires, demonstrating that phonon hydrodynamics is a powerful framework that can be applied in nanosystems even at room temperature.

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“…Hence a form of the jump conditions given in [1, Ch. 2, eqs [25][26][27] is easily retrieved. The momentum condition (9b) highlights a problem with this derivation form.…”

Section: Jump Conditions At the Interfacementioning

confidence: 99%

“…The Maxwell-Cattaneo equation gives rise to the hyperbolic or relativistic heat equation, which describes heat propagation with finite speed. Although the Guyer-Krumhansl equation was originally derived for situations with extremely low temperatures, recent studies have shown that it may be accurate at much higher temperatures [25]. It has been used to successfully predict the size dependence of the thermal conductivity in nanowires for temperatures ranging from 1 to 350 K [12,11], a feature that cannot be captured using Fourier's law.…”

Section: Non-fourier Heat Transfermentioning

confidence: 99%

The Stefan problem with variable thermophysical properties and phase change temperature

Myers

Hennessy

Calvo-Schwarzwälder

2020

International Journal of Heat and Mass Transfer

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In this paper we formulate a Stefan problem appropriate when the thermophysical properties are distinct in each phase and the phase-change temperature is size or velocity dependent. Thermophysical properties invariably take different values in different material phases but this is often ignored for mathematical simplicity. Size and velocity dependent phase change temperatures are often found at very short length scales, such as nanoparticle melting or dendrite formation; velocity dependence occurs in the solidification of supercooled melts. To illustrate the method we show how the governing equations may be applied to a standard one-dimensional problem and also the melting of a spherically symmetric nanoparticle. Errors which have propagated through the literature are highlighted. By writing the system in non-dimensional form we are able to study the large Stefan number formulation and an energy-conserving one-phase reduction. The results from the various simplifications and assumptions are compared with those from a finite difference numerical scheme. Finally, we briefly discuss the failure of Fourier's law at very small length and time-scales and provide an alternative formulation which takes into account the finite time of travel of heat carriers (phonons) and the mean free distance between collisions. * tmyers@crm.cat 1 arXiv:1904.05698v1 [physics.comp-ph]

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Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures

Cited by 86 publications

References 82 publications

Generalization of Fourier’s Law into Viscous Heat Equations

Generalization of Fourier’s Law into Viscous Heat Equations

Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics

The Stefan problem with variable thermophysical properties and phase change temperature

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