A moment model for phonon transport at room temperature

Abstract

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, its diameter depends on the temperature of the system. A simple model to describe phonon interaction with crystal boundary is employed to obtain macroscopic boundary conditions, where the reflection kernel is the superposition of diffusive reflection, specular reflection and isotropic scattering. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of 3 directional and 7 frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer, as well as Poiseuille flow of phonons. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained. * alirezam@uvic.ca 2

A General and Predictive Understanding of Thermal Transport from 1D- and 2D-Confined Nanostructures: Theory and Experiment

“…Hence, hydrodynamic effects can locally alter the heat transport even in semiconductors like silicon. In the limit where normal collisions dominate, Guyer and Krumhansl found α = 2; however, we use α = 1/3analogous to a fluid with zero volume viscosityin agreement with more recent works. ,,, To solve this equation, appropriate boundary conditions are implemented (see Methods and Supplementary Section 1). In addition, we require the thermal boundary resistance between the metal and the substrate.…”

Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures

“…2, which usually has recourse to numerical schemes such as the discrete ordinate method (DOM) [50] and Monte Carlo (MC) method [51]. It is a difficult task to develop a unified hydrodynamic model considering the phonon spectral properties which are diverse for different materials [39,40]. To capture the common features of non-Fourier heat conduction, the following assumptions are made as a first step [3]: (i) the isotropic assumption, in which phonon properties in one crystalline direction are representative of those in the whole wave vector space; (ii) the gray assumption, in which three identical acoustic phonon branches are considered with an effective constant relaxation time, with negligible contribution from the optical phonon branches; (iii) Debye's assumption, in which the linear phonon dispersion relation ω = v g k is adopted.…”

“…(29) is fully expressed as = v g τ R . Since it uses the same field variables as the traditional Fourier's description, the present hydrodynamic model avoids the complexity of classical moment methods involving the governing equation of higher-order moments [25,26,40]. Comparing to the C-V type heat transport Eq.…”

“…Furthermore, the derived nonequilibrium phonon distribution Eq. 31is dependent on the field variables and their gradients, in a different way from that as a function of local field variables in the classical Grad's type moment method [25,26,40,44]. In contrast to the second-or higher-order Chapman-Enskog expansion solution with higher-order gradient terms of field variables as a source of instability [52], the present closure method yields a nonequilibrium phonon distribution involving only first-order gradient terms ensuring a stable hydrodynamic equation.…”

“…This continuum model does not treat the kinetic effects in a spatially homogeneous material with temporal and spatial thermal variations at a characteristic dimension comparable to or even smaller than the phonon relaxation time and mean-free path [39]. The classical maximum entropy moment method is applied to derive macroscopic equations for phonon heat transport at room temperature in a recent study [40], which involves governing equations of at least 63 moments to reach satisfactory results. It will be inconvenient for practical applications due to the nontrivial solution of so many equations as well as the specification of both initial and boundary conditions for the higher-order moment variables.…”