A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales

A universal constitutive equation between the heat flux vector and the temperature gradient is proposed to cover the fundamental behaviors of diffusion (macroscopic in both space and time), wave (macroscopic in space but microscopic in time), phonon–electron interactions (microscopic in both space and time), and pure phonon scattering. The model is generalized from the dual-phase-lag concept accounting for the lagging behavior in the high-rate response. While the phase lag of the heat flux captures the small-scale response in time, the phase lag of the temperature gradient captures the small-scale response in space. The universal form of the energy equation facilitates identifications of the physical parameters governing the transition from one mechanism (such as diffusion or wave) to another (the phonon–electron interaction).