Dual-phase-lagging heat conduction based on Boltzmann transport equation

Abstract

In this work, the dual-phase-lagging model of the microscale heat conduction is re-derived analytically from the Boltzmann transport equation. Based on this model, the delay/advanced partial differential equations governing the microscale heat conduction are established. The method of separation of variables is applied to solve such delay/advanced partial differential equations. Finally, the oscillation of the microscale heat conduction is investigated.

Introduction

It has a long history to explore the macroscopic properties from the molecular level of description of the materials. This coarse-graining process allows us to bridge the gap between microscopic and meso/macroscopic descriptions. The understanding of coarse-graining and its correlated theories such as transport processes and irreversible phenomena was strengthened in the middle of last century by people like Kirkwood [12], Green [7], Kubo [15], Zwanzig [33], [34], [35] and many others. Recently, the rapid development of nano-technology gives a new impetus to this old topic. In the present work, we attempt to re-derive the dual-phase-lagging model of microscale heat conduction starting from the molecular level of description, the Boltzmann transport equation.

The fundamental law of the conventional heat conduction is the classical Fourier law

$$\mathbf{q}(\mathbf{r}\text{,}t)=-k\nabla T(\mathbf{r}\text{,}t)\text{,}$$

where the temperature gradient ∇T(r, t) is a vector function of the position vector r and the time variable t, the vector q(r, t) is called the heat flux, k is the thermal conductivity of the material which is a positive scalar quantity. This classical law has been widely and successfully applied in the conventional engineering heat conduction problems, in which the system has large spatial dimension and the emphasis is on the long time behavior. However, the Fourier law suffers some drawbacks. Firstly, it assumes the infinite speed of heat propagation, implying that a thermal disturbance applied at a certain location in a medium can be sensed immediately anywhere else in the medium. Secondly, because the heat flux and the temperature gradient are simultaneous, there is no difference between the cause and the effect of heat flow. This is doubted in the transient behavior at extremely short time, say, on the order of picoseconds to femtoseconds. An example is the ultrafast laser heating in thermal processing of materials. Thirdly, experimental observations of heat transport of the propagation of second sound, ballistic phonon propagation and phonon hydrodynamics in solids at low temperatures depart significantly from the usual parabolic description. Furthermore, due to the wide application of microdevices and the rapid development of modern microfabrication technology, more and more microdevices with micro and nano-scale dimension emerge in various micromechanical systems. It is well known that the conventional Fourier law leads to the unaccepted result for these microdevices [11].

Many efforts have been spent on the improvement of classical Fourier law. Cattaneo [3] and Vernotte [28], [29] proposed the CV model,

$$\tau \frac{\mathrm{\partial }\mathbf{q}}{\mathrm{\partial }t}+\mathbf{q}=-k\nabla T\text{,}$$

where τ is the time delay. This leads to a wave type of heat conduction equation called hyperbolic heat conduction equation [10]. The natural extension of CV model is

$$\mathbf{q}(\mathbf{r}\text{,}t+\tau )=-k\nabla T(\mathbf{r}\text{,}t)\text{,}$$

which was discussed in Tzou [19], [20], [21], [22], [23]. Further improvement of the model (3) leads to the dual-phase-lagging (DPL) model by Tzou [27]. It allows either the temperature gradient (cause) to precede the heat flux vector (effect) or the heat flux vector (cause) to precede the temperature gradient (effect) in transient processes. Mathematically, this is represented by [24], [25], [26], [27]

$$\mathbf{q}(\mathbf{r}\text{,}t+{\tau }_q)=-k\nabla T(\mathbf{r}\text{,}t+{\tau }_{\mathrm{T}})\text{,}$$

where the τ_T is the phase lag of the temperature gradient and the τ_q is the phase lag of the heat flux vector. For the case of τ_T > τ_q, the temperature gradient established across a material volume is a result of the heat flow, implying that the heat flux vector is the cause and the temperature gradient is the effect. For τ_T < τ_q, on the other hand, heat flow is developed by the temperature gradient established at an earlier time, implying that the temperature gradient is the cause. The first-order approximation of Eq. (4) reads

$$\mathbf{q}(\mathbf{r}\text{,}t)+{\tau }_q\frac{\mathrm{\partial }\mathbf{q}}{\mathrm{\partial }t}(\mathbf{r}\text{,}t)\cong -k\left\{\nabla T(\mathbf{r}\text{,}t)+{\tau }_{\mathrm{T}}\frac{\mathrm{\partial }}{\mathrm{\partial }t}[\nabla T(\mathbf{r}\text{,}t)]\right\}\text{.}$$

In the literatures, the dual-phase-lagging (abbreviated as DPL) model usually refers to this model. However, in the present paper, we mainly focus on the original DPL model expressed in (4).

Originally, the DPL heat conduction equation comes from the phonon–electron interaction model [17] and the phonon scattering model [8], [10]. These models have been developed in examining energy transport involving the high-rate heating in which the non-equilibrium thermodynamic transition and the microstructural effect become important associated with shortening of the response time. The high-rate heating is developing rapidly due to the advancement of high-power short-pulse laser technologies [1], [6], [13], [14], [16]. In addition to its application in the ultrafast pulse laser heating, the DPL heat conduction equation also arises in describing and predicting phenomena such as temperature pulses propagating in superfluid liquid helium, non-homogeneous lagging response in porous media, thermal lagging in amorphous materials, and effects of material defects and thermomechanical coupling etc. [27]. The study of the DPL heat conduction is thus of considerable importance in understanding and applying these rapidly emerging technologies. The well-posedness of DPL heat conduction was established [30], [31]. The thermal oscillation and resonance phenomena were investigated in detail by Xu and Wang [32] based on the approximate DPL model (5). Such phenomena are believed to be a manifestation of non-equilibrium behavior of microscale heat conduction. The Boltzmann transport equation (BTE), a theory of non-equilibrium heat and mass transport, may therefore be useful for examining the microscale heat conduction.

Indeed, the BTE is playing a central role in the study of microscale heat conduction. The classical Fourier law and the CV model for one-dimensional case was established from the BTE [18]. The phonon–electron interaction model [17] was derived from BTE on a quantum mechanical and statistical basis. Joshi and Majumdar [11] derived a phonon radiative transport equation between two parallel plates from the BTE for the heat transport in dielectric solid films. Based on the BTE, Chen [4], [5] proposed a ballistic–diffusive heat conduction equation of microscale heat transport in devices where the characteristic length is comparable to the mean free path of the energy-carrier and/or the characteristic time is comparable to the relaxation time of the energy-carrier. Encouraged by the successful applications of BTE in the microscale heat conduction, we attempt to establish the DPL model (4) from the BTE. The governing equation of DPL heat conduction, which is expressed as the delay/advanced partial differential equations, is then obtained by combining the DPL heat conduction model with the energy conservation equation. The associated initial conditions for this equation is prescribed. The method of separation of variables is employed to solve the DPL heat conduction problems. The oscillating features of microscale heat conduction are also investigated.