Phonon Hydrodynamics Research Articles

Nine-moment phonon hydrodynamics based on the maximum-entropy closure: one-dimensional flow

The maximum-entropy distribution functions applied to Callaway's model for phonon gas dynamics lead to a hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first member of this hierarchy of closures. Here emphasis is placed on analysing the next member, the 9-moment maximum-entropy system that involves the flux of the heat flux as an extra gas-state variable. After presenting a study of the one-dimensional, rotationally symmetric reduction of this system, we explicitly calculate a single generating function of three Lagrange multipliers in terms of which the reduced system of three evolution equations for these multipliers can be cast into a symmetric hyperbolic form. In the context of determining the Lagrange multipliers as explicit functions of the moment densities, we discuss new aspects of the expansion of various non-equilibrium quantities about quasi-equilibrium states. This expansion is fundamentally a non-equilibrium expansion that includes the heat flux in a non-perturbative manner, i.e., there are no unphysical limitations on the magnitude of the nonvanishing component of the heat flux to maintain a theory. Results are presented both at the first order in the expansion and at the second order. This enables us to verify the internal consistency of our approach and to justify the non-equilibrium generalization of the method of Grad.

Nine-moment phonon hydrodynamics based on the modified Grad-type approach: hyperbolicity of the one-dimensional flow

After expanding the distribution function about an anisotropic Planck function, the new moment closure method of Banach and Larecki applied to the Boltzmann–Peierls equation for the phonon gas dynamics leads to a whole hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first, non-perturbative member of this hierarchy of closures. In our previous paper (2004 J. Phys. A: Math. Gen. 37 9805), emphasis was placed on deriving the next member, the 9-moment anisotropic closure that involves the flux of the heat flux as an extra gas-state variable. Here, as a first step in effectively analysing this system, we present a study of the one-dimensional, rotationally symmetric reduction of these equations. Under the assumption of Callaway's model, a systematic procedure is derived which shows that the obtained system of three evolution equations for three nonvanishing gas-state variables can be cast into a symmetric hyperbolic form. For the sake of completeness, we describe explicitly the region of symmetric hyperbolicity in parameter space (the space defined by the gas-state variables). The evolution system is symmetric hyperbolic for significant ranges of physical conditions, i.e., there are effectively no unphysical limitations on the magnitude of the energy density and the heat flux. This paper also deals with the eigenvalue problem and calculates approximately the characteristic speeds.

Nine-moment phonon hydrodynamics based on the modified Grad-type approach: formulation

In this paper, we present the derivation of a new set of heat transport equations, called the equations of nine-moment phonon hydrodynamics, which are expected to describe transient processes under high thermal loads. The nine-moment model introduces the energy density, the heat flux and the flux of the heat flux as basic gas-state variables. The evolution equations for these variables are derived from the Grad-type expansion method applied to the Boltzmann–Peierls equation with Callaway's collisional terms. The basic idea is to expand the phase density about an anisotropic Planck distribution. The advantage of using this distribution is that the heat flux is incorporated into the model in a non-perturbative manner, thereby allowing virtually arbitrarily large values for the components of the heat flux. Special emphasis is placed on finding explicit closed-form expressions for the moment flux and collisional quantities in terms of independent gas-state variables. Our model involves two relaxation times and it seems particularly suited for describing phonon flows in the regime where the relaxation time for normal processes is much smaller than the relaxation time for resistive processes.

Phonon modes and heat capacity of monolayer films adsorbed in spherical pores

We examine the hydrodynamic phonon spectrum of a monolayer film adsorbed on the wall of a spherical pore. Due to the boundary conditions, the monolayer film exhibits a discrete phonon spectrum. The corresponding density of states per unit frequency is thus a set of delta functions, and the heat capacity exhibits Arrhenius behavior at low temperatures. At high temperatures, the heat capacity approaches the C approximately T2 behavior of a two-dimensional monolayer film. Our results for the spherical surface film are compared to previous calculations for films confined to a cylindrical surface.

The role of long-wave longitudinal phonons in kinetics of insulators

It is shown that long-wave longitudinal phonons considerably affect heat transfer in most of existing crystals in spite of symmetry degeneracy of transverse vibrational branches. This is because of close values of phase velocity of transverse phonon modes in these crystals. Such effects are most pronounced in phonon hydrodynamics and in the propagation of second sound. It is shown that the thermal conductivity of bulk samples is a nonmonotonic function of the parameter characterizing the difference between the velocities of transverse modes.

Phonon gas: A lattice Boltzmann description.

A lattice Boltzmann computational scheme is developed and applied to a model of the phonon gas. This model has two types of phonons, transverse and longitudinal, that interact with one another through a three-phonon collision process. Exact and relaxation time approximation treatments of the collision operator are described. The computational scheme is tested in a study of the propagation of small-amplitude second sound where the predictions of linearized phonon hydrodynamics are confirmed. It is then employed to study two problems of current interest in nonlinear phonon hydrodynamics: (1) the propagation of linear second sound in the presence of a steady heat current, and (2) the propagation of nonlinear second sound.

Irreducible tensor description. III. Thermodynamics of a low-temperature phonon gas

Let one assume that the interacting phonon gas, whose behavior is governed by the Boltzmann–Peierls equation, inhabits an insulating crystal at sufficiently low temperature. Then, within the framework of a single acoustic phonon branch and of an isotropic long-wavelength approximation to the dispersion relation, the simplest acceptable version of extended irreversible thermodynamics, based upon the nine-moment system of differential equations for the slow and fast gas-state variables, is carefully investigated. It is clearly demonstrated that, in virtue of the structure simplifications just mentioned, the conceptually different (macroscopic, kinetic, and variational) procedures, which are discussed in this paper, appear complementary to each other. Finally, with the help of a suitable contraction of the nine-moment system of field equations, for Callaway’s relaxation model a slightly generalized nonlinear variant of ordinary low-temperature phonon hydrodynamics is explicitly derived.

Phonon hydrodynamics in solids

In this lecture we shall deal with hydrodynamic motion of phonons in solids, i. e. with situations where the phonon gas can flow or carry out density oscillations like an ordinary fluid.

Green’s function approach to phonon hydrodynamics in solids

In the first part the Green’s function approach developed byKadanoff andBaym is used to derive a transport equation for phonons in a dielectric anharmonic crystal. The approximations which reduce this generalized equation to the Peierls equation are exhibited. In the second part, neglecting Umklapp processes, the linearized equation is applied to hydrodynamic phenomena. In a system close to local thermal equilibrium second sound and Poiseuille flow may exist. The damping through normal processes is approximately calculated by a momentum-dependent relaxation time. The coupling between the phonon and the dilatation fields is discussed.

Thermal Conductivity, Second Sound, and Phonon Hydrodynamic Phenomena in Nonmetallic Crystals

A variety of phonon-gas phenomena in nonmetals are discussed in a unified manner using a set of macroscopic equations developed from the solution of the linearized phonon Boltzmann equation. This set of macroscopic equations, appropriate for the description of a low-temperature phonon gas, is solved for a cylindrical sample in the limit ${\ensuremath{\lambda}}_{N}\ensuremath{\ll}R$; ${\ensuremath{\lambda}}_{N}\ensuremath{\lambda}_{R}^{}{}_{}{}^{z}\ensuremath{\gg}{R}^{2}$. Here ${\ensuremath{\lambda}}_{N}$ is the normal-process mean free path, $\ensuremath{\lambda}_{R}^{}{}_{}{}^{z}$ is the mean free path for momentum-loss scattering calculated in the Ziman limit, and $R$ is the radius of the sample. The solution in this limit exhibits Poiseuille flow of the phonon gas as first discussed by Sussmann and Thellung. An equation for the thermal conductivity which correctly includes this phenomenon is found. Using this equation, the possible outcomes of steady-state thermal-conductivity measurements are discussed in terms of the microscopic scattering rates. Heat-pulse propagation is discussed from a similar point of view. The existence of Poiseuille flow in steady-state thermal-conductivity measurements bears directly on the possibility of observing second sound in solids. A quantitative analysis of available data on LiF suggests that the chemical purity of these samples sets very stringent limits on the observation of either of these effects. The observation of Poiseuille flow in solid ${\mathrm{He}}^{4}$ samples by Mezov-Deglin strongly suggests that this material is a prime subject for investigations of second-sound propagation.