Da Yu Tzou

On the Wave Theory in Heat Conduction

This work contains three major components: a thorough review on the research emphasizing engineering applications of the thermal wave theory, special features in thermal wave propagation, and the thermal wave model in relation to the microscopic two-step model. For the sake of convenience, the research works are classified according to their individual emphases. Special features in thermal wave propagation include the sharp wavefront and rate effects, the thermal shock phenomenon, the thermal resonance phenomenon, and reflections and refractions of thermal waves across a material interface. By employing the dual-phase-lag concept, we show that the energy equation may be reduced to that governing the heat transport through the metal lattice in the microscopic two-step model

The generalized lagging response in small-scale and high-rate heating

The generalized lagging behavior in solids under high-rate heating is derived by precise correlation with the hyperbolic two-step model. The ballistic behavior of heat transport in the electron gas is found to be captured by the second-order effect of the phase lag of the heat flux vector. In contrast to the parabolic two-step model, the ballistic behavior results in a sharp wavefront in the history of heat propagation. The analytical expression for the thermal wave speed is derived. In comparison with the classical diffusion and the wave models, the two phase lags in the lagging response result in a much deeper thermal penetration depth and much higher temperature in the heat-affected zone.

Experimental support for the lagging behavior in heat propagation

This work examines the lagging behavior for heat transport in small-scale and fast-transient processes. The experimental result by Qiu et al. for the femtosecond transient response in gold films and that by Bertman and Sandiford for the temperature pulse traveling through superfluid liquid helium are re-examined with emphasis on the lagging behavior. The model employing the phase-lag concept provides as competent or even better results when describing the special response observed in these experiments.

Temperature-dependent thermal lagging in ultrafast laser heating

Abstract Temperature-dependent phase-lags are incorporated in the dual-phase-lag (DPL) model to fully describe the experimental data of femtosecond (fs) laser heating on gold films of various thicknesses in the sub-micron range. An explicit finite difference algorithm is developed to perform the nonlinear analysis, which recovers the Crank–Nicholson stability criterion in the special case of Fourier diffusion. The exponents in the temperature-dependent thermal properties are determined by minimizing the mean error between the numerical and the experimental results. The lagging model with temperature-dependent thermal properties enables a consistent description of all the available experimental data for ultrafast laser heating on gold films.

Hot-electron blast induced by ultrashort-pulsed lasers in layered media

Femtosecond laser heating on metals produces a blasting force in the sub-picosecond domain, which exerts on the metal lattices along with the non-equilibrium heat flow from hot electrons. Such a hot-electron blast depends on both temperature and temperature gradient in the electron gas, resulting in pronounced effects in multi-layered metal films due to discontinuous heat transfer and load transmission across the interface. This work employs the parabolic two-step model to study the effect of the hot-electron blast in multi-layered thin metal films. Dominating physical parameters are identified to characterize the ultrafast heating and deformation across the interface.

An engineering assessment to the relaxation time in thermal wave propagation

Abstract The physical significance of the relaxation time in the wave theory of heat conduction is further studied in this work. Thermodynamically, it is confirmed that the relaxation time results from the rate equation within the mainframe of the second law in the nonequilibrium, irreversible thermodynamics. Mechanically, on the other hand, the relaxation time results from the phase-lag between the heat flux vector and the temperature gradient in a high-rate response. In transition from a diffusion behavior to the wave propagation, lastly, the relaxation time is found to be the physical instant at which the intrinsic length scales merge with each other.

Shock wave formation around a moving heat source in a solid with finite speed of heat propagation

Abstract The thermal field around a moving heat source in a solid with finite speed of heat propagation is studied analytically in this work. A thermal Mach number M defined as the ratio between the speed of the moving heat source and that of the heat propagation in the solid is introduced in the analysis. The resulting energy equation is found to be elliptic, parabolic, and hyperbolic in the subsonic ( M M = 1), and supersonic ( M > 1) ranges, respectively. Thermal shock wave is shown to exist in the physical domain as the speed of the moving heat source is equal to or faster than that of the heat propagation, and the thermal shock angle is obtained analytically as sin −1 ( 1 M ) for M ⩾ 1. In the numerical examples, the evolution of the temperature and the heat flux distributions in the heat affected zone is present as a function of the thermal Mach number and a swinging phenomenon for the thermal field in transition is discussed.

Thermal shock waves induced by a moving crack : a heat flux formulation

Abstract The thermal shock phenomena induced by a rapidly propagating crack tip in a solid medium was studied recently via the temperature formulation ( ASME J. Heat Transfer 112 , 21–27 (1990)). In order to further confirm the unique features obtained in the previous research for the thermal waves emanating from the crack tip, the present work employs the flux formulation and further examines the same physical phenomena from a different viewpoint. The thermal shock discontinuities result from the energy accumulation in a preferential direction as the speed of the propagating crack tip exceeds the heat propagation speed in the solid. By employing the flux formulation in the thermal wave model, the present work investigates the thermal shock formation, the evolutions of the heat affected zone and the thermally undisturbed zone, and the transition of the r -dependency of the heat flux vector in transition of the thermal Mach number from the subsonic to the supersonic ranges. The analogy between the temperature and the heat flux in the near-tip region is established and the thermal shock phenomena are confirmed from a theoretical point of view.

Thermal Lagging in Random Media

The lagging behavior of heat transport in amorphous materials is studied by the use of temperature formulation along with surface heating. The percolating network is alternatively viewed as the physical mechanism causing delayed responses between the heat flux vector and temperature gradient in transporting heat. Experimental results of microsecond responses in rough carbon samples and submillisecond responses in weakly bonded copper spheres are re-examined with emphasis on the possible thermal lagging. Picosecond anomalous diffusion predicted by the fractal and fracton models are compared for silica aerogels and silicon dioxides. It has been shown that the fractal behavior in space can be interpreted in terms of the lagging behavior in time. Nomenclature

Ultrafast Deformation in Femtosecond Laser Heating

The hot-electron blasting model is extended in this work to describe the ultrafast deformation in thin metal films during the sub-picosecond to picosecond domain. The driving force exerting on the cold metal lattices is induced by the highly heated electrons, dictated by both the temperature and temperature gradient established in the hot electron during the picosecond transient. Since the metal lattices remain almost thermally undisturbed in this highly non-equilibrium regime, the resulting ultrafast deformation patterns cannot be described by the classical dynamical theory of thermoelasticity. The phonon-electron interaction model is used to describe the electron temperature and hence the driving force. The dominating parameters characterizing the nonlinearly coupled ultrafast heating and deformation are identified. Method of lines is used to solve the coupled field equations describing ultrafast deformation in the picosecond domain