Derivation of stretched exponential tap density equations of granular powders

The tap density of granular powders was found to be better fitted with the stretched exponential law. In our previous work, the stretched exponential tap density equations were derived with the rate process theory and free volume concept, under the assumption that the particle packing rate during the tapping process obeys the stretched Arrhenius equation, which, however, has an empirical origin. In this article, the above assumption is eliminated and attempts are made to obtain the stretched exponential tap density equations from very fundamental bases. In a vertical tapping process, the probability of particles attaining certain energy states is assumed to obey the Boltzmann distribution and particles traveling from one site to another are assumed to follow a very common memoryless random exponential law. The stretched exponential tap density equations are thus derived and all parameters acquire clear physical meanings. The most important parameter, the stretched exponential, is demonstrated to correlate with the interparticle forces: a small value may indicate a strong adhesive or cohesive interaction. Therefore, the stretched exponential could be a better indicator for powder flowability correlated with particle interactions as well.