Lagging Behavior in other Transport Processes

Abstract

Lagging behavior is not a prerogative of heat transport during the ultrafast transient. For the transport processes involving microstructural interactions such as mass interdiffusion of different species, chemical reactions, and thermoelectrical coupling, the lagging behavior is expected as well, in times comparable to the characteristic times describing the microstructural interaction effects. This chapter extends the same phase-lag concept in Fourier's law to Fick's law for mass diffusion. Experimental results for the thin-film growth of silicon dioxide and the time evolution of intermetallic compound layers for a wide variety of electronic materials are examined to identify the sources of lagging. Threshold values for the phase lag of the density gradient and phase lag of the mass flux vector are determined from the experimental curves, which are now in hours and days. Coupling between thermal and electrical fields follows, with emphasis on extracting the lagging behavior during the ultrafast transient across the mushy zone. Resemblance between the dual-phase-lag heat-transfer model and viscoelasticity/viscoelastic fluids, as well as its perfect correlations to the transient response in nanofluids are derived. Regardless of different sources for the delayed responses, therefore, the lagging behavior seems to be common during the ultrafast transient in different fields.Lagging response results from the finite times required to turn a cause into an effect. Such finite times are characterized by the two phase lags, τ T and τ q , in the dual-phase-lag model, where heat flux and temperature gradient can be either the cause or the effect in heat flow. The resulting delayed response intrinsically alters the nature of heat transport, in times comparable to the phase lags. Beyond the scope of microscale heat transfer, the delayed response between the cause and effect should be a general concern for transient processes in other disciplines in engineering and science. It may not have been evident in the past because the process time may be much longer than the phase lags characterizing the delayed response. As the response time continuously shortens into the pico/femtosecond domain and the physical scale shrinks into the region of nanometers, however, the lagging behavior may need to be re-evaluated in examining what we have studied before.This chapter is dedicated to the lagging behavior in mass transport. Experimental results for the thin-film growth of silicon dioxide and growth of intermetallic compound layers in solder