Thermal Wave Based on the Thermomass Model

Abstract

In times comparable to the characteristic time of the energy carriers, Fourier’s law of heat conduction breaks down and heat may propagate as waves. Based on the concept of thermomass, which is defined as the equivalent mass of phonon gas in dielectrics, according to the Einstein’s mass-energy relation, the phonon gas in the dielectrics is described as a weighty, compressible fluid. Newton mechanics has been applied to establish the equation of state and the equation of motion for the phonon gas as in fluid mechanics, because the drift velocity of a phonon gas is normally much less than the speed of light. The propagation velocity of the thermal wave in the phonon gas is derived directly from the equation of state for the phonon gas, rather than from the relaxation time in the Cattaneo–Vernotte (CV) model (Cattaneo, C., 1948, “Sulla Conduzione Del Calore,” Atti Semin. Mat. Fis. Univ. Modena, 3, pp. 83–101; Vernotte, P., 1958, “Paradoxes in the Continuous Theory of the Heat Equation,” C. R. Acad. Bulg. Sci., 246, pp. 3154–3155). The equation of motion for the phonon gas gives rise to the thermomass model, which depicts the general relation between the temperature gradient and heat flux. The linearized conservation equations for the phonon gas lead to a damped thermal wave equation, which is similar to the CV-wave equation, but with different characteristic time. The lagging time in the resulting thermal wave equation is related to the wave velocity in the phonon gas, which is approximately two orders of magnitude larger than the relaxation time adopted in the CV-wave model for the lattices. A numerical example for fast transient heat conduction in a silicon film is presented to show that the temperature peaks resulting from the thermomass model are much higher than those resulting from the CV-wave model. Due to the slower thermal wave velocity in the phonon gas, by as much as one order of magnitude, the damage due to temperature overshooting may be more severe than that expected from the CV-wave model.