Decomposition model for phonon thermal conductivity of a monatomic lattice

Abstract

An analytical treatment of decomposition of the phonon thermal conductivity of a crystal with a monatomic unit cell is developed on the basis of a twostage decay of the heat current autocorrelation function observed in molecular dynamics simulations. It is demonstrated that the contributions from the acoustic short-and long-range phonon modes to the total phonon thermal conductivity can be presented in the form of simple kinetic formulas, consisting of products of the heat capacity and the average relaxation time of the considered phonon modes as well as the square of the average phonon velocity. On the basis of molecular dynamics calculations of the heat current autocorrelation function, this treatment allows for a self-consistent numerical evaluation of the aforementioned variables. In addition, the presented analysis allows, within the Debye approximation, for the identification of the temperature range where classical molecular dynamics simulations can be employed for the prediction of phonon thermal transport properties. As a case example, Cu is considered. IntroductionIt is well known that the thermodynamic and transport properties of a crystal lattice can be generally described using the concept of phonons (lattice vibrations or lattice waves) [1][2][3]. For example, the temperature dependence of the lattice heat capacity can be well accounted for within the harmonic approximation of the lattice vibrations by the Debye model [1]. In particular, the Debye model, in accordance with experiment, predicts that at high temperatures T [ T D (T D ¼ hx D =k B is the Debye temperature, x D is the Debye frequency -the highest allowed phonon frequency in the crystal, ℏ is the Planck constant divided by 2π, and k B is the Boltzmann constant), the lattice heat capacity can be approximated by the classical value C % 3Nk B =V (N is the number of atoms in the crystal volume V), which is known as Dulong and Petit value; whilst at low temperatures T \T D the lattice heat capacity starts to decrease (following C ∼ T 3 law at T ( T D ) which is a reflection of quantum effects on the phonon population [1].

Two-fluid nature of phonon heat conduction in a monatomic lattice

“…It was generally noted [2][3][4][5][6][7] that the second term in Equation (4), which is related to the second stage of the HCACF relaxation, dominates and contains most of the temperature dependence of the phonon thermal conductivity (see, as an example, Figure 2 with related data for the MD models of f.c.c. Cu and Ni [5][6][7]).…”

“…In the past, equilibrium molecular dynamics (MD) simulations in conjunction with the Green-Kubo formalism have been successfully used on several occasions to calculate the lattice (phonon) thermal conductivity of crystals with a monatomic unit cell [1][2][3][4][5][6][7]. All of them have revealed a two-stage decay (due to phonon-phonon interactions) in the heat current autocorrelation function (HCACF).…”

“…Taking into account the observed shapes of the HCACF of different MD models of monatomic f.c.c. crystals investigated at different temperatures, we have recently found and argued [5][6][7] that the HCACF of a monatomic lattice can be nicely and universally modelled with the following analytical function of the form:…”

“…The five parameters (A 1 , A 2 , τ 1 , τ 2 and x c ) of Equation (1) can readily be found by the fitting of C JJ t ð Þ extracted from MD simulations [5][6][7] (see, as an example, Figure 1 with the related data for the MD models of f.c.c. Cu and Ni calculated with embedded atom method potentials [12] and [13], respectively).…”

“…Hence, the analytical model for the HCACF given by Equation (1) suggests that k can be represented as [5][6][7]:…”