An enhanced Gray model for nondiffusive heat conduction solved by implicit lattice Boltzmann method

Many studies have revealed that Fourier's law becomes invalid in the study of heat transfer in micro/nanoscale systems due to phonon-mediated nondiffusive heat transport. It is necessary to develop high-fidelity non-Fourier models and numerical methods to study nondiffusive heat transfer for both experimental data analysis in nanothermometry and thermal design in micro-/nano-systems. Starting from the phonon Boltzmann transport equation (BTE), we develop an enhanced Gray (EG) model by considering the second-order terms in Taylor expansion. In the proposed enhanced Gray BTE (EG-BTE), two parameters associated with inherent material properties, i.e., the diffusive relaxation time and the ballistic mean free time, are used to characterize the diffusive heat conduction and nondiffusive heat conduction, respectively. While the diffusive relaxation time is calculated directly from the thermal conductivity, the ballistic mean free time of nonmetallic materials can be determined from transient grating experiments. An implicit lattice Boltzmann method (ILBM) is developed to solve the EG-BTE, which is unconditionally stable. The EG-BTE model is validated by comparing predictions of thermal decay processes with transient grating experiments. After being validated, the EG-BTE model is applied to study heat conduction across thin films from ballistic to diffusive regime at room temperature as well as thermal wave propagation in crystal at low temperatures. The numerical results demonstrate that the EG-BTE provides a unified model to describe multiscale heat conduction covering both nondiffusive and diffusive regimes.