A mathematical analysis of milling mechanics in a planetary ball mill

A detailed mathematical analysis is presented to advance the current understanding of the mechanics of milling operation in a planetary ball mill in terms of a global Cartesian reference space. The ab initio calculations have identified the role of milling parameters in determining the condition of detachment of the ball from the vial wall. The condition of an 'effective' impact has been identified in terms of the vial-to-disk speed ratio. It emerges from the analysis that the role of velocity components of the ball at the instant of its impact on the vial wall warrants proper consideration, because the tangential force determines the lower bound of the vial-to-disk speed ratio conducive for effective transfer of impact energy to the powder charge in the mill. Finally, a comparison of the experimental results of grain size reduction during milling of elemental Fe and Cu-Al powder blend with the similar predictions of the present analysis demonstrates for the first time that elastic properties of the balls and vials play an important role in determining the rate of structural refinement during ball milling.