Solution of the Linearized Phonon Boltzmann Equation

Guyer, R. A.; Krumhansl, J. A.

doi:10.1103/physrev.148.766

Cited by 781 publications

(499 citation statements)

References 18 publications

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“…The width and length of the sample are assumed to be much larger than the phonon mean free path. work, we demonstrate from first-principles calculations that a displaced distribution of phonons can occur in graphene even in the presence of weak R-scattering, which validates the assumption of the displaced distribution used in the past studies 12,13 and correspondingly shows hydrodynamic phonon transport in graphene. The recent progress in calculating the phonon-phonon scattering matrix using first-principles 14,15 and numerical methods for solving the Peierls-Boltzmann equation 16,17 enable the calculation of phonon distributions in a specific material under a temperature gradient.…”

Section: Displaced Distribution Functionsupporting

confidence: 84%

“…The displaced distribution we present here is the first explicit confirmation of the assumption used in past theoretical studies regarding hydrodynamic phonon transport 12,13,25,26 . It also provides a theoretical basis that we will exploit to examine phonon Poiseuille flow and second sound in the following sections, further highlighting the significant hydrodynamic phonon transport in graphene.…”

Section: Displaced Distribution Functionsupporting

confidence: 81%

“…2, we show this phonon distribution in an infinitely large graphene sheet with 0.1% concentration of isotope 13 C at 100 K under a static temperature gradient. The temperature gradient is required to maintain the drift motion since R-scattering cannot be completely ignored in real materials.…”

Section: Displaced Distribution Functionmentioning

confidence: 71%

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Hydrodynamic phonon transport in suspended graphene

et al. 2015

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Recent studies of thermal transport in nanomaterials have demonstrated the breakdown of Fourier's law through observations of ballistic transport. Despite its unique features, another instance of the breakdown of Fourier's law, hydrodynamic phonon transport, has drawn less attention because it has been observed only at extremely low temperatures and narrow temperature ranges in bulk materials. Here, we predict on the basis of first-principles calculations that the hydrodynamic phonon transport can occur in suspended graphene at significantly higher temperatures and wider temperature ranges than in bulk materials. The hydrodynamic transport is demonstrated through drift motion of phonons, phonon Poiseuille flow and second sound. The significant hydrodynamic phonon transport in graphene is associated with graphene's two-dimensional features. This work opens a new avenue for understanding and manipulating heat flow in two-dimensional materials.

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Section: Displaced Distribution Functionsupporting

confidence: 84%

Section: Displaced Distribution Functionsupporting

confidence: 81%

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Hydrodynamic phonon transport in suspended graphene

et al. 2015

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“…All these models consider the heat carriers as a fluid, whose hydrodynamic-like equations of motion describe the heat transport. The phonon hydrodynamics lays on the Guyer-Krumhansl transport equation [16][17][18][19][20][21][22] for the heat flux q, i.e., (1) τq…”

Section: Heat Transport Equations Beyond the Fourier Law And Hydrodynmentioning

confidence: 99%

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

Jou¹,

Cimmelli

2016

Communications in Applied and Industrial Mathematics

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“…Letting and ℓ tend to zero, one finds back Fouriers' law = − ∇ . With ℓ = 0 and omitting the second order derivatives in space and time, one recovers Cattaneo's relation, while setting only = 0, expression (18) reduces to GuyerKrumhansl's relation [9]. It may be convenient to reformulate the above results in dimensionless form.…”

Section: A Generalized Transport Equation In Terms Of the Heat Fluxmentioning

confidence: 99%

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

Machrafi

Lebon

2016

Applied Mathematics Letters

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A generalized heat transport equation applicable at small length and short time scales is proposed. It is based on extended irreversible thermodynamics with an infinite number of highorder heat fluxes selected as state variables. Extensions of Fick's and Ohm's laws are also formulated. As a numerical illustration, heat conduction in a rigid body subject to fixed and oscillatory temperature boundary conditions is discussed. KeywordsBallistic heat transport; Small length-systems; High-frequency; Extended irreversible thermodynamics IntroductionThe increasing interest in nano-technology has led to new insights in the study of heat transport. It is well known that heat transfer at micro and nanoscale behaves differently from that at macroscales [1,2]. At small length scales, transport of heat in complex systems is best quantified by means of the so-called Knudsen number ≡ ℓ/ with ℓ denoting the mean free path of the heat carriers, namely phonons, and , the characteristic dimension of the system under study. The Knudsen number becomes typically comparable or larger than one for micro-and nano-systems, in which case heat transport is referred to as ballistic, i.e. dominated by phonon collisions with the walls. If the Knudsen number is much smaller than one, heat transport is simply diffusive, i.e. dominated by phonon-phonon collisions inside the system and described by Fourier's law. For small-length systems, as well as for high-frequency processes, Fourier's law is no longer valid. Another drawback associated to Fourier's law is that it implies that temperature spreads infinitely fast through the whole body, which is physically unsustainable. These observations justify the need to generalize Fourier's law. This can be achieved by various ways, for instance, via the resolution of Boltzmann's equation [3], making use of the dual time approach [4] or by computer simulations [5]. Here we follow a different route based on Extended Irreversible Thermodynamics (EIT) [6] whose main characteristic is to upgrade the heat flux and higher order heat fluxes to the rank of independent variables. In most applications, for convenience, the analysis is restricted by taking heat flux as single extra variable. In this letter, we go one step further by selecting an infinite number of higher order fluxes. The use of a large number of flux variables finds its justification in the recent progress of nanotechnology and high-frequency processes. The procedure described in this paper is by no means limited to heat transfer but can be easily generalized to other transport phenomena, as electrical conduction and matter diffusion. As a numerical application, we consider the problem of heat conduction in a one-dimensional rigid body of length L at rest, subject to two different kinds of boundary conditions, made explicit in Section 5. The theoretical model for the general heat transport equation is presented in the next Section 2.

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Solution of the Linearized Phonon Boltzmann Equation

Cited by 781 publications

References 18 publications

Hydrodynamic phonon transport in suspended graphene

Hydrodynamic phonon transport in suspended graphene

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

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